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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">V</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><br/>  for <font color="Olive" title="b6">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S1</font><br/>  for <font color="Olive" title="b7">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">S2</font>  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="b5">V</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b6">A</font>,<font color="Olive" title="b7">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <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">V</font> <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; <font color="Olive" title="b3">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">A</font> <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>  holds <br/><span class="p1"> { <span class="default"> <font color="Olive" title="b8">E</font> where <font color="Olive" title="b8">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="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>)</span>)</span> :  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b4">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b8">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b5">V</font></span>)</span>,<font color="Olive" title="b3">M1</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> <span class="p1">(<span class="default"><font color="Olive" title="b8">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b5">V</font></span>)</span> </span> } </span>  is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <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><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">V</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><br/>  for <font color="Olive" title="b5">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b6">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S2</font>  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="b4">V</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b5">A</font>,<font color="Olive" title="b6">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <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">V</font> <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; <font color="Olive" title="b2">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">A</font> <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>  holds <br/><span class="p1"> { <span class="default"> <font color="Olive" title="b7">E</font> where <font color="Olive" title="b7">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="Olive" title="b1">S2</font>)</span>)</span> :  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b3">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b4">V</font></span>)</span>,<font color="Olive" title="b2">M1</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> <span class="p1">(<span class="default"><font color="Olive" title="b7">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b4">V</font></span>)</span> </span> } </span>  is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <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><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">V</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><br/>  for <font color="Olive" title="b4">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b5">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font>  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="b3">V</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b4">A</font>,<font color="Olive" title="b5">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <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">V</font> <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; <font color="Olive" title="b1">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">A</font> <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>  holds <br/><span class="p1"> { <span class="default"> <font color="Olive" title="b6">E</font> where <font color="Olive" title="b6">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="Olive" title="b2">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b6">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b3">V</font></span>)</span>,<font color="Olive" title="b1">M1</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> <span class="p1">(<span class="default"><font color="Olive" title="b6">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b3">V</font></span>)</span> </span> } </span>  is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <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><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">V</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><br/>  for <font color="Olive" title="b3">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b4">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font>  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="b2">V</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <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">V</font> <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; <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">A</font> <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>  holds <br/><span class="p1"> { <span class="default"> <font color="Olive" title="b5">E</font> where <font color="Olive" title="b5">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="Olive" title="b1">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b5">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b2">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <span class="p1">(<span class="default"><font color="Olive" title="b5">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b2">V</font></span>)</span> </span> } </span>  is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <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><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">V</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><br/>  for <font color="Olive" title="b2">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b3">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font>  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">V</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <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">V</font> <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; <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">A</font> <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>  holds <br/><span class="p1"> { <span class="default"> <font color="Olive" title="b4">E</font> where <font color="Olive" title="b4">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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b4">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b1">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="b4">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b1">V</font></span>)</span> </span> } </span>  is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <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><br/><span class="kw">let </span><font color="Maroon" title="c7">V</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">  for <font color="Olive" title="b1">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font>  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="c7">V</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="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> &amp; <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">V</font> <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; <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">A</font> <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>  holds <br/><span class="p1"> { <span class="default"> <font color="Olive" title="b3">E</font> where <font color="Olive" title="b3">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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b3">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="b3">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <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><br/><span class="kw">let </span><font color="Maroon" title="c8">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</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">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font>  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="c7">V</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c8">A</font>,<font color="Olive" title="b1">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <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">V</font> <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; <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">A</font> <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>  holds <br/><span class="p1"> { <span class="default"> <font color="Olive" title="b2">E</font> where <font color="Olive" title="b2">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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b2">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="b2">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <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><br/><span class="kw">let </span><font color="Maroon" title="c9">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</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"> ( <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="c7">V</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <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">V</font> <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; <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">A</font> <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>  implies <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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <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><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:142"/><span class="lab"><font color="Green" title="E107">A01</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="E2:142"/><span class="lab"><font color="Green" title="E108">A02</font></span>: 
<font color="Maroon" title="c7">V</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c9">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:142"/><span class="lab"><font color="Green" title="E109">A0</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> <font color="Maroon" title="c7">V</font> <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">and </span><br/><a NAME="E4:142"/><span class="lab"><font color="Green" title="E110">PS2</font></span>: 
<font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">A</font> <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> 
 ; <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="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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <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><br/>

<span class="kw">set </span><font color="Maroon" title="c10">Z</font> = <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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">XX2</font> = <font color="Maroon" title="c2">X2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</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/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="142_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">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">E</font> where <font color="Olive" title="b2">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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b2">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="b2">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">A</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="c12">A</font> <a href="hidden.html#R2" title="HIDDEN: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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  implies <font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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><br/><span class="kw">assume </span><a NAME="E1:142_1"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN: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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</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"> <font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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><br/><a NAME="E2:142_1"/><span class="kw">then </span>
 ex <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 <br/>( <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> )
 ;<br/><span class="kw">hence </span><a NAME="E3:142_1"/>
<font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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>
 ; <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="E7:142"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A1</font></span>: 
<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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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>
 
;<br/>
<a NAME="E8:142"/>
 for <font color="Olive" title="b1">A1</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <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>  st <font color="Olive" title="b1">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>  &amp;  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">E</font> where <font color="Olive" title="b2">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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b2">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="b2">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  holds <br/> <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">A1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">E</font> where <font color="Olive" title="b2">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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Olive" title="b2">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="b2">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="142_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <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>; <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">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>  &amp;  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c12">A1</font> <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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  implies  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">A1</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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:142_2"/><span class="lab"><font color="Green" title="E112">A2</font></span>: 
( <font color="Maroon" title="c12">A1</font> is  <a href="setlim_1.html#V1" title="SETLIM_1:attr.1">monotone</a>  &amp;  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c12">A1</font> <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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</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">  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">A1</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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span> </span><br/>

<a NAME="E2:142_2"/><span class="lab"><font color="Green" title="E113">A4</font></span>: 
 for <font color="Olive" title="b1">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">V</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c12">A1</font> holds <br/><font color="Olive" title="b1">V</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="142_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">W</font> be    <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"> ( <font color="Maroon" title="c13">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c12">A1</font> implies <font color="Maroon" title="c13">W</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="E1:142_2_1"/>
<font color="Maroon" title="c13">W</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c12">A1</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="c13">W</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="E2:142_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">W</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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_2"><span class="lab"><font color="Green" title="E112">A2</font></span></a></span>;<br/>
<a NAME="E3:142_2_1"/><span class="kw">then </span>
 ex <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 <br/>( <font color="Maroon" title="c13">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E4:142_2_1"/>
<font color="Maroon" title="c13">W</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>
<a NAME="E3:142_2"/>
 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="c12">A1</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="142_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">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="c12">A1</font> <a href="prob_2.html#K1" title="PROB_2: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/>

<a NAME="E1:142_2_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">A1</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/>
<a NAME="E2:142_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">n</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> <font color="Maroon" title="c12">A1</font>
 
<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="E3:142_2_2"/>
<font color="Maroon" title="c12">A1</font> <a href="prob_2.html#K1" title="PROB_2: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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142_2"><span class="lab"><font color="Green" title="E113">A4</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</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="c13">A2</font> = <font color="Maroon" title="c12">A1</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="E5:142_2"/><span class="lab"><font color="Green" title="E114">PP</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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="142_2_3"><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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span></span><br/>

<a NAME="E1:142_2_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c13">A2</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/>
<a NAME="E2:142_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">n</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> <font color="Maroon" title="c13">A2</font>
 
<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/>
<a NAME="E3:142_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">A2</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> <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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_2"><span class="lab"><font color="Green" title="E112">A2</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E4:142_2_3"/><span class="kw">then </span>
 ex <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 <br/>( <font color="Maroon" title="c13">A2</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> <font color="Olive" title="b1">E</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:142_2_3"/>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</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>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_2_4"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:142_2_4"/>
( <font color="Maroon" title="c12">A1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a>  or <font color="Maroon" title="c12">A1</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_2"><span class="lab"><font color="Green" title="E112">A2</font></span></a>, <a class="ref" href="setlim_1.html#D1" onmouseover="rs('setlim_1/D1')" onmouseout="rh()">SETLIM_1:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_2_4_1"><span class="kw">suppose </span></a><a NAME="E1:142_2_4_1"/><span class="lab"><font color="Green" title="E115">A3</font></span>: 
<font color="Maroon" title="c12">A1</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</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="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">A1</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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span> </span><br/><div class="add"><a NAME="E2:142_2_4_1"/>
 <a href="measure4.html#K2" title="MEASURE4:func.2">union</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c12">A1</font></span>)</span> <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="#E2:142_2"><span class="lab"><font color="Green" title="E113">A4</font></span></a>, <a class="ref" href="measure1.html#T35" onmouseover="rs('measure1/T35')" onmouseout="rh()">MEASURE1:35</a></span>;<br/><a NAME="E3:142_2_4_1"/><span class="kw">then </span>
 <a href="measur11.html#K3" title="MEASUR11:func.3">Union</a> <font color="Maroon" title="c12">A1</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="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">E</font> =  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">A1</font> 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="txt" href="#E1:142_2_4_1"><span class="lab"><font color="Green" title="E115">A3</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">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> $1</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>);<br/><a NAME="E5:142_2_4_1"/><span class="lab"><font color="Green" title="E116">T1</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="c2">X2</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="142_2_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">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="c2">X2</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="c15">n</font>,<font color="Olive" title="b1">f</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">f1</font> =  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="c2">X2</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="c16">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="c15">n</font>,<font color="Maroon" title="c16">f1</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:142_2_4_1_1"/>
<font color="Maroon" title="c16">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="c15">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="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:142_2_4_1"/><span class="lab"><font color="Green" title="E117">T2</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="c15">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="#E5:142_2_4_1"><span class="lab"><font color="Green" title="E116">T1</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c16">F</font> = <font color="Maroon" title="c15">F</font> as   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><a NAME="E9:142_2_4_1"/><span class="lab"><font color="Green" title="E118">T2a</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="c16">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="142_2_4_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span><br/>

<a NAME="E1:142_2_4_1_2"/>
<font color="Maroon" title="c17">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="E2:142_2_4_1_2"/>
<font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:142_2_4_1"><span class="lab"><font color="Green" title="E117">T2</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="E10:142_2_4_1"/>
<font color="Maroon" title="c16">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:142_2_4_1"><span class="lab"><font color="Green" title="E117">T2</font></span></a></span>;<br/><a NAME="E11:142_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">T3</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="c16">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="c11">XX2</font> &amp; <font color="Maroon" title="c16">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="E12:142_2_4_1"/><span class="lab"><font color="Green" title="E120">T4</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="c2">X2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c16">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="142_2_4_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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="c2">X2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="c18">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font></span><br/>



<a NAME="E1:142_2_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">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="E2:142_2_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="#E9:142_2_4_1"><span class="lab"><font color="Green" title="E118">T2a</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="E13:142_2_4_1"/><span class="lab"><font color="Green" title="E121">T5</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="c16">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="c16">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="142_2_4_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">n</font>, <font color="Maroon" title="c18">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m</font></span>)</span></span><br/>

<a NAME="E1:142_2_4_1_4"/>
( <font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>) &amp; <font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">m</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:142_2_4_1"><span class="lab"><font color="Green" title="E118">T2a</font></span></a></span>;<br/>
<a NAME="E2:142_2_4_1_4"/><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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">XX2</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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">XX2</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="E3:142_2_4_1_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">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="E14:142_2_4_1"/><span class="lab"><font color="Green" title="E122">T6</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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c11">XX2</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="142_2_4_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> is <font color="Maroon" title="c11">XX2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/>

<a NAME="E1:142_2_4_1_5"/>
<font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:142_2_4_1"><span class="lab"><font color="Green" title="E118">T2a</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:142_2_4_1_5"/>
<font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> is <font color="Maroon" title="c11">XX2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142"><span class="lab"><font color="Green" title="E107">A01</font></span></a>, <a class="ref" href="measur11.html#D14" target="_self" onmouseover="rs('measur11/D14')" onmouseout="rh()">DefXvol</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="E15:142_2_4_1"/><span class="lab"><font color="Green" title="E123">T7</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="c2">X2</font>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c16">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="142_2_4_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">n</font>, <font color="Maroon" title="c18">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="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">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="c2">X2</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">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="E1:142_2_4_1_6"/><span class="lab"><font color="Green" title="E124">T71</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">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="c2">X2</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">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/>

<div><span class="kw">hereby </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>
<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="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"> ( <font color="Maroon" title="c19">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m</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="E1:142_2_4_1_6_1"/>
<font color="Maroon" title="c19">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m</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="E2:142_2_4_1_6_1"/><span class="lab"><font color="Green" title="E125">T72</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">m</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_2_4_1"><span class="lab"><font color="Green" title="E115">A3</font></span></a>, <a class="txt" href="#E1:142_2_4_1_6"><span class="lab"><font color="Green" title="E124">T71</font></span></a>, <a class="ref" href="prob_1.html#D5" onmouseover="rs('prob_1/D5')" onmouseout="rh()">PROB_1:def 5</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="E126">T73</font></span>: <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"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c19">x</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="#E1:142"><span class="lab"><font color="Green" title="E107">A01</font></span></a>, <a class="ref" href="measur11.html#D14" target="_self" onmouseover="rs('measur11/D14')" onmouseout="rh()">DefXvol</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:142_2_4_1"><span class="lab"><font color="Green" title="E120">T4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">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/><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"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">m</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c19">x</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">m</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="#E1:142"><span class="lab"><font color="Green" title="E107">A01</font></span></a>, <a class="ref" href="measur11.html#D14" target="_self" onmouseover="rs('measur11/D14')" onmouseout="rh()">DefXvol</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:142_2_4_1"><span class="lab"><font color="Green" title="E120">T4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m</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="E5:142_2_4_1_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m</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="#E2:142_2_4_1_6_1"><span class="lab"><font color="Green" title="E125">T72</font></span></a>, <a class="txt" href="#E3:142_2_4_1_6_1"><span class="lab"><font color="Green" title="E126">T73</font></span></a>, <a class="ref" href="measur11.html#T21" target="_self" onmouseover="rs('measur11/T21')" onmouseout="rh()">Th15</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>


</div><span class="kw">end;</span></div><a NAME="E16:142_2_4_1"/><span class="lab"><font color="Green" title="E124">T8</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="c2">X2</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</font> holds <br/><font color="Maroon" title="c16">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="142_2_4_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <font color="Maroon" title="c17">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</font> implies <font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:142_2_4_1_7"/>
<font color="Maroon" title="c17">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</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="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">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="142_2_4_1_7_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="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">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="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">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="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="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">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="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:142_2_4_1_7_1"/>
<font color="Maroon" title="c19">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">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="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">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="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">n</font></span><br/><a NAME="E2:142_2_4_1_7_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">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="c16">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="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:142_2_4_1"><span class="lab"><font color="Green" title="E123">T7</font></span></a></span>;<br/><a NAME="E3:142_2_4_1_7_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">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="c16">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="c17">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="E4:142_2_4_1_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">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="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">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="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="E3:142_2_4_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">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="E4:142_2_4_1_7"/>
<font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c17">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="c17">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a><span class="kw"> such that </span><br/><a NAME="E18:142_2_4_1"/><span class="lab"><font color="Green" title="E125">V2</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="c17">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="c6">M2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">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="c6">M2</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c16">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="c17">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:142_2_4_1"><span class="lab"><font color="Green" title="E119">T3</font></span></a>, <a class="txt" href="#E13:142_2_4_1"><span class="lab"><font color="Green" title="E121">T5</font></span></a>, <a class="txt" href="#E14:142_2_4_1"><span class="lab"><font color="Green" title="E122">T6</font></span></a>, <a class="txt" href="#E15:142_2_4_1"><span class="lab"><font color="Green" title="E123">T7</font></span></a>, <a class="txt" href="#E16:142_2_4_1"><span class="lab"><font color="Green" title="E124">T8</font></span></a>, <a class="ref" href="mesfunc9.html#T52" onmouseover="rs('mesfunc9/T52')" onmouseout="rh()">MESFUNC9:52</a>, <a class="ref" href="mesfunc8.html#D2" onmouseover="rs('mesfunc8/D2')" onmouseout="rh()">MESFUNC8:def 2</a></span>;<br/><a NAME="E19:142_2_4_1"/>
 <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="c16">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="c16">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="E20:142_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">V4</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="c16">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:142_2_4_1"><span class="lab"><font color="Green" title="E119">T3</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="E21:142_2_4_1"/>
 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="c2">X2</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="c16">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="c16">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="142_2_4_1_8"><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="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"> ( <font color="Maroon" title="c18">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="c16">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="c16">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:142_2_4_1_8"/>
<font color="Maroon" title="c18">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="c16">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="c16">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font></span><br/>

<a NAME="E2:142_2_4_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">L2</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c16">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">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="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">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="c19">G</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c3">S1</font><span class="kw"> such that </span><br/><a NAME="E4:142_2_4_1_8"/><span class="lab"><font color="Green" title="E128">L3</font></span>: 
( <font color="Maroon" title="c19">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="c19">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#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c18">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c18">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c19">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c14">E</font>,<font color="Maroon" title="c18">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c18">x</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_2_4_1"><span class="lab"><font color="Green" title="E115">A3</font></span></a>, <a class="ref" href="measur11.html#T109" target="_self" onmouseover="rs('measur11/T109')" onmouseout="rh()">Th109</a></span>;<br/>
<a NAME="E5:142_2_4_1_8"/>
 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="c16">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> <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="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c19">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="142_2_4_1_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">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="c16">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">n</font> <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="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c19">G</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">n</font></span><br/>

<a NAME="E1:142_2_4_1_8_1"/><span class="lab"><font color="Green" title="E129">L5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">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="E130">L4</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">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">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c20">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="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="#E7:142_2_4_1"><span class="lab"><font color="Green" title="E117">T2</font></span></a></span>
<br/>.= 
<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"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c18">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142"><span class="lab"><font color="Green" title="E107">A01</font></span></a>, <a class="ref" href="measur11.html#D14" target="_self" onmouseover="rs('measur11/D14')" onmouseout="rh()">DefXvol</a></span>
;<br/>
<a NAME="E3:142_2_4_1_8_1"/>
 <a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c18">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">n</font></span>)</span>,<font color="Maroon" title="c18">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c18">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="E4:142_2_4_1_8_1"/><span class="kw">then </span>
 <a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c18">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c20">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:142_2_4_1_8"><span class="lab"><font color="Green" title="E128">L3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:142_2_4_1_8_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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">n</font> <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="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c19">G</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142_2_4_1_8_1"><span class="lab"><font color="Green" title="E130">L4</font></span></a>, <a class="txt" href="#E1:142_2_4_1_8_1"><span class="lab"><font color="Green" title="E129">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="E6:142_2_4_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c18">x</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">M1</font> <a href="measure8.html#K11" title="MEASURE8:func.11">*</a> <font color="Maroon" title="c19">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="E7:142_2_4_1_8"/><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="c16">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c14">E</font>,<font color="Maroon" title="c18">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c18">x</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142_2_4_1_8"><span class="lab"><font color="Green" title="E127">L2</font></span></a>, <a class="txt" href="#E4:142_2_4_1_8"><span class="lab"><font color="Green" title="E128">L3</font></span></a>, <a class="ref" href="measure8.html#T26" onmouseover="rs('measure8/T26')" onmouseout="rh()">MEASURE8:26</a></span>;<br/>
<a NAME="E8:142_2_4_1_8"/><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="c16">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c18">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/>
<span class="kw">hence </span><a NAME="E9:142_2_4_1_8"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c16">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="#E1:142"><span class="lab"><font color="Green" title="E107">A01</font></span></a>, <a class="ref" href="measur11.html#D14" target="_self" onmouseover="rs('measur11/D14')" onmouseout="rh()">DefXvol</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="E22:142_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">V3</font></span>: 
 <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c16">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:142_2_4_1"><span class="lab"><font color="Green" title="E126">V4</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="c18">J</font> = <font color="Maroon" title="c7">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</font>;<br/><a NAME="E23:142_2_4_1"/><span class="lab"><font color="Green" title="E128">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">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</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="E24:142_2_4_1"/>
 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">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</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="142_2_4_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">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="c19">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">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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="E1:142_2_4_1_9"/>
<font color="Maroon" title="c19">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">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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="c20">n1</font> = <font color="Maroon" title="c19">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="E3:142_2_4_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">n1</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</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="E4:142_2_4_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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="c19">J</font> = <font color="Maroon" title="c7">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</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="#E23:142_2_4_1"><span class="lab"><font color="Green" title="E128">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="E26:142_2_4_1"/><span class="lab"><font color="Green" title="E129">R11</font></span>: 
<font color="Maroon" title="c19">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="#E1:142_2_4_1"><span class="lab"><font color="Green" title="E115">A3</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="E27:142_2_4_1"/>
<font color="Maroon" title="c13">A2</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="#E1:142_2_4_1"><span class="lab"><font color="Green" title="E115">A3</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="E28:142_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">R13</font></span>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c19">J</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font>
 <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/><a NAME="E29:142_2_4_1"/><span class="lab"><font color="Green" title="E131">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="E30:142_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">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="E31:142_2_4_1"/>
 <a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c19">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="E32:142_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">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="c19">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="c19">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:142_2_4_1"><span class="lab"><font color="Green" title="E132">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="E33:142_2_4_1"/>
 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="c17">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="c19">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="142_2_4_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">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="c17">I</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">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="c19">J</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">n</font></span><br/>

<a NAME="E1:142_2_4_1_10"/><span class="lab"><font color="Green" title="E134">R21</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">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="c17">I</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">n</font> = 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c20">n</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:142_2_4_1"><span class="lab"><font color="Green" title="E125">V2</font></span></a></span>
<br/>.= 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:142_2_4_1"><span class="lab"><font color="Green" title="E117">T2</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="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:142_2"><span class="lab"><font color="Green" title="E114">PP</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="c19">J</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c20">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="E3:142_2_4_1_10"/>
<font color="Maroon" title="c17">I</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">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="c19">J</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c20">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:142_2_4_1"><span class="lab"><font color="Green" title="E133">R2</font></span></a>, <a class="txt" href="#E1:142_2_4_1_10"><span class="lab"><font color="Green" title="E134">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="E34:142_2_4_1"/><span class="kw">then </span>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c17">I</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"><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="c19">J</font></span>)</span>
 <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="E35:142_2_4_1"/><span class="kw">then </span>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c17">I</font> <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="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:142_2_4_1"><span class="lab"><font color="Green" title="E130">R13</font></span></a>, <a class="txt" href="#E26:142_2_4_1"><span class="lab"><font color="Green" title="E129">R11</font></span></a>, <a class="txt" href="#E32:142_2_4_1"><span class="lab"><font color="Green" title="E133">R2</font></span></a>, <a class="txt" href="#E29:142_2_4_1"><span class="lab"><font color="Green" title="E131">R3</font></span></a>, <a class="ref" href="measure8.html#T26" onmouseover="rs('measure8/T26')" onmouseout="rh()">MEASURE8:26</a></span>;<br/><span class="kw">hence </span><a NAME="E36:142_2_4_1"/>
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">A1</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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:142_2_4_1"><span class="lab"><font color="Green" title="E125">V2</font></span></a>, <a class="txt" href="#E22:142_2_4_1"><span class="lab"><font color="Green" title="E127">V3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_2_4_2"><span class="kw">suppose </span></a><a NAME="E1:142_2_4_2"/><span class="lab"><font color="Green" title="E115">A3</font></span>: 
<font color="Maroon" title="c12">A1</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</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="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">A1</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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span> </span><br/><div class="add"><a NAME="E2:142_2_4_2"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c12">A1</font></span>)</span> <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="#E2:142_2"><span class="lab"><font color="Green" title="E113">A4</font></span></a>, <a class="ref" href="measure1.html#T35" onmouseover="rs('measure1/T35')" onmouseout="rh()">MEASURE1:35</a></span>;<br/><a NAME="E3:142_2_4_2"/><span class="kw">then </span>
 <a href="prob_1.html#K3" title="PROB_1:func.3">Intersection</a> <font color="Maroon" title="c12">A1</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="setlim_1.html#T8" onmouseover="rs('setlim_1/T8')" onmouseout="rh()">SETLIM_1:8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">E</font> =  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">A1</font> 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="txt" href="#E1:142_2_4_2"><span class="lab"><font color="Green" title="E115">A3</font></span></a>, <a class="ref" href="setlim_1.html#T64" onmouseover="rs('setlim_1/T64')" onmouseout="rh()">SETLIM_1:64</a></span>;<br/><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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> $1</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>);<br/><a NAME="E5:142_2_4_2"/><span class="lab"><font color="Green" title="E116">T1</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="c2">X2</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="142_2_4_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">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="c2">X2</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="c15">n</font>,<font color="Olive" title="b1">f</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">f1</font> =  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="c2">X2</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="c16">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="c15">n</font>,<font color="Maroon" title="c16">f1</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:142_2_4_2_1"/>
<font color="Maroon" title="c16">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="c15">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="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:142_2_4_2"/><span class="lab"><font color="Green" title="E117">T2</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="c15">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="#E5:142_2_4_2"><span class="lab"><font color="Green" title="E116">T1</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c16">F</font> = <font color="Maroon" title="c15">F</font> as   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><a NAME="E9:142_2_4_2"/><span class="lab"><font color="Green" title="E118">T2a</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="c16">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="142_2_4_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span><br/>

<a NAME="E1:142_2_4_2_2"/>
<font color="Maroon" title="c17">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="E2:142_2_4_2_2"/>
<font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:142_2_4_2"><span class="lab"><font color="Green" title="E117">T2</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="E10:142_2_4_2"/>
<font color="Maroon" title="c16">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:142_2_4_2"><span class="lab"><font color="Green" title="E117">T2</font></span></a></span>;<br/><a NAME="E11:142_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">T3</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="c16">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="c11">XX2</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/><a NAME="E12:142_2_4_2"/><span class="lab"><font color="Green" title="E120">T4</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="c2">X2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c16">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="142_2_4_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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="c2">X2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="c18">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font></span><br/>



<a NAME="E1:142_2_4_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">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="E2:142_2_4_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="#E9:142_2_4_2"><span class="lab"><font color="Green" title="E118">T2a</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="E13:142_2_4_2"/>
 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="c16">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="c16">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="142_2_4_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">n</font>, <font color="Maroon" title="c18">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m</font></span>)</span></span><br/>

<a NAME="E1:142_2_4_2_4"/>
( <font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>) &amp; <font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">m</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:142_2_4_2"><span class="lab"><font color="Green" title="E118">T2a</font></span></a></span>;<br/>
<a NAME="E2:142_2_4_2_4"/><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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">XX2</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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">XX2</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="E3:142_2_4_2_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c17">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="c16">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">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><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">F</font> = <font color="Maroon" title="c16">F</font> as   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc8.html#D2" onmouseover="rs('mesfunc8/D2')" onmouseout="rh()">MESFUNC8:def 2</a></span>;<br/><a NAME="E15:142_2_4_2"/><span class="lab"><font color="Green" title="E121">T6</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c17">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; <font color="Maroon" title="c17">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c11">XX2</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="142_2_4_2_5"><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="c17">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; <font color="Maroon" title="c17">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c11">XX2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )</span><br/>

<a NAME="E1:142_2_4_2_5"/>
<font color="Maroon" title="c17">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:142_2_4_2"><span class="lab"><font color="Green" title="E118">T2a</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:142_2_4_2_5"/>
( <font color="Maroon" title="c17">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>  &amp; <font color="Maroon" title="c17">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c11">XX2</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142"><span class="lab"><font color="Green" title="E107">A01</font></span></a>, <a class="ref" href="measur11.html#D14" target="_self" onmouseover="rs('measur11/D14')" onmouseout="rh()">DefXvol</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="E16:142_2_4_2"/><span class="lab"><font color="Green" title="E122">T7</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="c2">X2</font><br/>  for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b2">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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">F</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b3">m</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="142_2_4_2_6"><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="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">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</font> &amp; <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/><span class="p1">(<span class="default"><font color="Maroon" title="c17">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="Maroon" title="c18">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">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="Maroon" title="c18">x</font></span><br/><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="c18">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</font> &amp; <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">m</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c17">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> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">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> )</span><br/>



<span class="kw">assume </span><a NAME="E1:142_2_4_2_6"/>
( <font color="Maroon" title="c18">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">XX2</font> &amp; <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c17">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> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">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></span><br/>

<a NAME="E2:142_2_4_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">T72</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_2_4_2"><span class="lab"><font color="Green" title="E115">A3</font></span></a>, <a class="ref" href="prob_1.html#D4" onmouseover="rs('prob_1/D4')" onmouseout="rh()">PROB_1:def 4</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="E124">T73</font></span>: <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"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c18">x</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c19">n</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="#E1:142"><span class="lab"><font color="Green" title="E107">A01</font></span></a>, <a class="ref" href="measur11.html#D14" target="_self" onmouseover="rs('measur11/D14')" onmouseout="rh()">DefXvol</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">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="txt" href="#E12:142_2_4_2"><span class="lab"><font color="Green" title="E120">T4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">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/>
<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"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c18">x</font>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c20">m</font></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="#E1:142"><span class="lab"><font color="Green" title="E107">A01</font></span></a>, <a class="ref" href="measur11.html#D14" target="_self" onmouseover="rs('measur11/D14')" onmouseout="rh()">DefXvol</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:142_2_4_2"><span class="lab"><font color="Green" title="E120">T4</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">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>
<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="E5:142_2_4_2_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">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="c17">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="#E2:142_2_4_2_6"><span class="lab"><font color="Green" title="E123">T72</font></span></a>, <a class="txt" href="#E3:142_2_4_2_6"><span class="lab"><font color="Green" title="E124">T73</font></span></a>, <a class="ref" href="measur11.html#T21" target="_self" onmouseover="rs('measur11/T21')" onmouseout="rh()">Th15</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="E17:142_2_4_2"/><span class="lab"><font color="Green" title="E123">M3</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="E18:142_2_4_2"/>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">XX2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:142_2_4_2"><span class="lab"><font color="Green" title="E118">T2a</font></span></a></span>;<br/><a NAME="E19:142_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">M1</font></span>: 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">XX2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:142_2"><span class="lab"><font color="Green" title="E114">PP</font></span></a></span>;<br/><a NAME="E20:142_2_4_2"/>
<span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:142_2_4_2"><span class="lab"><font color="Green" title="E123">M3</font></span></a>, <a class="ref" href="measure1.html#T31" onmouseover="rs('measure1/T31')" onmouseout="rh()">MEASURE1:31</a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/><a NAME="E21:142_2_4_2"/><span class="kw">then </span>
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">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="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">XX2</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="#E3:142"><span class="lab"><font color="Green" title="E109">A0</font></span></a>, <a class="txt" href="#E19:142_2_4_2"><span class="lab"><font color="Green" title="E124">M1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><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="E23:142_2_4_2"/><span class="lab"><font color="Green" title="E125">V2</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="c6">M2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">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#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c18">I</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="c6">M2</font>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c17">F</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:142_2_4_2"><span class="lab"><font color="Green" title="E119">T3</font></span></a>, <a class="txt" href="#E15:142_2_4_2"><span class="lab"><font color="Green" title="E121">T6</font></span></a>, <a class="txt" href="#E16:142_2_4_2"><span class="lab"><font color="Green" title="E122">T7</font></span></a>, <a class="ref" href="mesfun10.html#T18" onmouseover="rs('mesfun10/T18')" onmouseout="rh()">MESFUN10:18</a></span>;<br/><a NAME="E24:142_2_4_2"/>
 <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="c17">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="c17">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="E25:142_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">V4</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="c17">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:142_2_4_2"><span class="lab"><font color="Green" title="E119">T3</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="E26:142_2_4_2"/>
 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="c2">X2</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="c17">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="c17">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="142_2_4_2_7"><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="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"> ( <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="c17">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="c17">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="E1:142_2_4_2_7"/>
<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="c17">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="c17">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="E2:142_2_4_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">L2</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c17">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="c17">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="c3">S1</font><span class="kw"> such that </span><br/><a NAME="E4:142_2_4_2_7"/><span class="lab"><font color="Green" title="E128">L3</font></span>: 
( <font color="Maroon" title="c20">G</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</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#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</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#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</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#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c14">E</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#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c19">x</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_2_4_2"><span class="lab"><font color="Green" title="E115">A3</font></span></a>, <a class="ref" href="measur11.html#T111" target="_self" onmouseover="rs('measur11/T111')" onmouseout="rh()">Th111</a></span>;<br/>
<a NAME="E5:142_2_4_2_7"/>
<font color="Maroon" title="c20">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></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#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c19">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:142_2_4_2_7"><span class="lab"><font color="Green" title="E128">L3</font></span></a></span>;<br/>
<a NAME="E6:142_2_4_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">L31</font></span>: 
<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="c20">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c19">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T31" onmouseover="rs('measure1/T31')" onmouseout="rh()">MEASURE1:31</a>, <a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E7:142_2_4_2_7"/>
 <a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c19">x</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142"><span class="lab"><font color="Green" title="E108">A02</font></span></a>, <a class="ref" href="measur11.html#T22" target="_self" onmouseover="rs('measur11/T22')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E8:142_2_4_2_7"/><span class="kw">then </span>
<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"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c19">x</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T31" onmouseover="rs('measure1/T31')" onmouseout="rh()">MEASURE1:31</a></span>;<br/>
<a NAME="E9:142_2_4_2_7"/><span class="kw">then </span>
<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="c20">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:142_2_4_2_7"><span class="lab"><font color="Green" title="E129">L31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:142_2_4_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">LL</font></span>: 
<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="c20">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></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="#E4:142"><span class="lab"><font color="Green" title="E110">PS2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:142_2_4_2_7"/>
 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="c17">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="c5">M1</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="142_2_4_2_7_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="c17">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="c5">M1</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="E1:142_2_4_2_7_1"/><span class="lab"><font color="Green" title="E131">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="E132">L4</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c17">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="c17">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</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">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="#E7:142_2_4_2"><span class="lab"><font color="Green" title="E117">T2</font></span></a></span>
<br/>.= 
<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"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</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">V</font></span>)</span>,<font color="Maroon" title="c19">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142"><span class="lab"><font color="Green" title="E107">A01</font></span></a>, <a class="ref" href="measur11.html#D14" target="_self" onmouseover="rs('measur11/D14')" onmouseout="rh()">DefXvol</a></span>
;<br/>
<a NAME="E3:142_2_4_2_7_1"/>
 <a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</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">V</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#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</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#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</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="E4:142_2_4_2_7_1"/><span class="kw">then </span>
 <a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">A2</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">V</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="#E4:142_2_4_2_7"><span class="lab"><font color="Green" title="E128">L3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:142_2_4_2_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">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="c5">M1</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="#E2:142_2_4_2_7_1"><span class="lab"><font color="Green" title="E132">L4</font></span></a>, <a class="txt" href="#E1:142_2_4_2_7_1"><span class="lab"><font color="Green" title="E131">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="E12:142_2_4_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c17">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="c5">M1</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="E13:142_2_4_2_7"/><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="c17">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="c5">M1</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#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c14">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#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c19">x</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142_2_4_2_7"><span class="lab"><font color="Green" title="E127">L2</font></span></a>, <a class="txt" href="#E4:142_2_4_2_7"><span class="lab"><font color="Green" title="E128">L3</font></span></a>, <a class="txt" href="#E10:142_2_4_2_7"><span class="lab"><font color="Green" title="E130">LL</font></span></a>, <a class="ref" href="measure8.html#T31" onmouseover="rs('measure8/T31')" onmouseout="rh()">MEASURE8:31</a></span>;<br/>
<a NAME="E14:142_2_4_2_7"/><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="c17">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="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</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/>
<span class="kw">hence </span><a NAME="E15:142_2_4_2_7"/>
<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c17">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#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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="#E1:142"><span class="lab"><font color="Green" title="E107">A01</font></span></a>, <a class="ref" href="measur11.html#D14" target="_self" onmouseover="rs('measur11/D14')" onmouseout="rh()">DefXvol</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="E27:142_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">V3</font></span>: 
 <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c17">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:142_2_4_2"><span class="lab"><font color="Green" title="E126">V4</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">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</font>;<br/><a NAME="E28:142_2_4_2"/><span class="lab"><font color="Green" title="E128">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">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</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="E29:142_2_4_2"/>
 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">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</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="142_2_4_2_8"><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">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</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="E1:142_2_4_2_8"/>
<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">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</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="E3:142_2_4_2_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</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="c13">A2</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">V</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="E4:142_2_4_2_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</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">V</font> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c13">A2</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="#E28:142_2_4_2"><span class="lab"><font color="Green" title="E128">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="E31:142_2_4_2"/><span class="lab"><font color="Green" title="E129">R11</font></span>: 
<font color="Maroon" title="c20">J</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:142_2_4_2"><span class="lab"><font color="Green" title="E115">A3</font></span></a>, <a class="ref" href="setlim_2.html#T21" onmouseover="rs('setlim_2/T21')" onmouseout="rh()">SETLIM_2:21</a></span>;<br/><a NAME="E32:142_2_4_2"/>
<font color="Maroon" title="c13">A2</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="#E1:142_2_4_2"><span class="lab"><font color="Green" title="E115">A3</font></span></a>, <a class="ref" href="setlim_1.html#T64" onmouseover="rs('setlim_1/T64')" onmouseout="rh()">SETLIM_1:64</a></span>;<br/><a NAME="E33:142_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">R13</font></span>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c20">J</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font>
 <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/><a NAME="E34:142_2_4_2"/><span class="lab"><font color="Green" title="E131">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="E35:142_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">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="E36:142_2_4_2"/>
 <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="E37:142_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">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="#E35:142_2_4_2"><span class="lab"><font color="Green" title="E132">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="E38:142_2_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">A2</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/><a NAME="E39:142_2_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">J</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">V</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/><a NAME="E40:142_2_4_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">J</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:142_2_4_2"><span class="lab"><font color="Green" title="E131">R3</font></span></a>, <a class="ref" href="measure1.html#T31" onmouseover="rs('measure1/T31')" onmouseout="rh()">MEASURE1:31</a></span>;<br/><a NAME="E41:142_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">K1</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="c20">J</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></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="#E3:142"><span class="lab"><font color="Green" title="E109">A0</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E42:142_2_4_2"/>
 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="142_2_4_2_9"><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="E1:142_2_4_2_9"/><span class="lab"><font color="Green" title="E135">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="c6">M2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">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="#E23:142_2_4_2"><span class="lab"><font color="Green" title="E125">V2</font></span></a></span>
<br/>.= 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-vol</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">A2</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">V</font></span>)</span>,<font color="Maroon" title="c5">M1</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:142_2_4_2"><span class="lab"><font color="Green" title="E117">T2</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="c13">A2</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">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:142_2"><span class="lab"><font color="Green" title="E114">PP</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="E3:142_2_4_2_9"/>
<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="#E37:142_2_4_2"><span class="lab"><font color="Green" title="E133">R2</font></span></a>, <a class="txt" href="#E1:142_2_4_2_9"><span class="lab"><font color="Green" title="E135">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="E43:142_2_4_2"/><span class="kw">then </span>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c18">I</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"><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>
 <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="E44:142_2_4_2"/><span class="kw">then </span>
 <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Maroon" title="c18">I</font> <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="c14">E</font> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c7">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:142_2_4_2"><span class="lab"><font color="Green" title="E130">R13</font></span></a>, <a class="txt" href="#E31:142_2_4_2"><span class="lab"><font color="Green" title="E129">R11</font></span></a>, <a class="txt" href="#E37:142_2_4_2"><span class="lab"><font color="Green" title="E133">R2</font></span></a>, <a class="txt" href="#E34:142_2_4_2"><span class="lab"><font color="Green" title="E131">R3</font></span></a>, <a class="txt" href="#E41:142_2_4_2"><span class="lab"><font color="Green" title="E134">K1</font></span></a>, <a class="ref" href="measure8.html#T31" onmouseover="rs('measure8/T31')" onmouseout="rh()">MEASURE8:31</a></span>;<br/><span class="kw">hence </span><a NAME="E45:142_2_4_2"/>
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">A1</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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:142_2_4_2"><span class="lab"><font color="Green" title="E125">V2</font></span></a>, <a class="txt" href="#E27:142_2_4_2"><span class="lab"><font color="Green" title="E127">V3</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:142"/>
<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="c6">M2</font>,<span class="p1">(<span class="default"><a href="measur11.html#K20" title="MEASUR11:func.20">X-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> <font color="Maroon" title="c7">V</font></span>)</span>,<font color="Maroon" title="c5">M1</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> <font color="Maroon" title="c7">V</font></span>)</span> </span> } </span>  is   <a href="prob_3.html#NM2" title="PROB_3:NM.2">MonotoneClass</a> of <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="#E7:142"><span class="lab"><font color="Green" title="E111">A1</font></span></a>, <a class="ref" href="prob_3.html#T69" onmouseover="rs('prob_3/T69')" onmouseout="rh()">PROB_3:69</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/>


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