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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">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S1</font>  st <font color="Olive" title="b3">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">B</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/> <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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>)</span>)</span> :  ex <font color="Olive" title="b6">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b7">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b6">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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="Olive" title="b5">E</font>,<font color="Olive" title="b7">y</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b4">B</font></span>)</span> ) &amp; (  for <font color="Olive" title="b7">V</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">S2</font> holds  <font color="Olive" title="b6">F</font> is <font color="Olive" title="b7">V</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) </span> } </span> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">S1</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b2">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b3">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>  st <font color="Olive" title="b2">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">B</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/> <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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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="Olive" title="b1">S2</font>)</span>)</span> :  ex <font color="Olive" title="b5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b6">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">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="Olive" title="b4">E</font>,<font color="Olive" title="b6">y</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b3">B</font></span>)</span> ) &amp; (  for <font color="Olive" title="b6">V</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S2</font> holds  <font color="Olive" title="b5">F</font> is <font color="Olive" title="b6">V</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) </span> } </span> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">S2</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>  st <font color="Olive" title="b1">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">B</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/> <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="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> :  ex <font color="Olive" title="b4">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b5">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b4">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b3">E</font>,<font color="Olive" title="b5">y</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b2">B</font></span>)</span> ) &amp; (  for <font color="Olive" title="b5">V</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <font color="Olive" title="b4">F</font> is <font color="Olive" title="b5">V</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) </span> } </span> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">M1</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>  st <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">B</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/> <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="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> :  ex <font color="Olive" title="b3">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b4">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b3">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">y</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="Olive" title="b2">E</font>,<font color="Olive" title="b4">y</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b1">B</font></span>)</span> ) &amp; (  for <font color="Olive" title="b4">V</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <font color="Olive" title="b3">F</font> is <font color="Olive" title="b4">V</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) </span> } </span> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">B</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"> ( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c6">B</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  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> :  ex <font color="Olive" title="b2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b2">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</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="Olive" title="b1">E</font>,<font color="Olive" title="b3">y</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> ) &amp; (  for <font color="Olive" title="b3">V</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <font color="Olive" title="b2">F</font> is <font color="Olive" title="b3">V</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) </span> } </span>  )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c7">K</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> :  ex <font color="Olive" title="b2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b2">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</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="Olive" title="b1">E</font>,<font color="Olive" title="b3">y</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> ) &amp; (  for <font color="Olive" title="b3">V</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <font color="Olive" title="b2">F</font> is <font color="Olive" title="b3">V</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E1:117"/>
<font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c6">B</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">  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> :  ex <font color="Olive" title="b2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b2">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</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="Olive" title="b1">E</font>,<font color="Olive" title="b3">y</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> ) &amp; (  for <font color="Olive" title="b3">V</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <font color="Olive" title="b2">F</font> is <font color="Olive" title="b3">V</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) </span> } </span> </span><br/>

<a NAME="E2:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">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> :  ex <font color="Olive" title="b2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b2">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</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="Olive" title="b1">E</font>,<font color="Olive" title="b3">y</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> ) &amp; (  for <font color="Olive" title="b3">V</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <font color="Olive" title="b2">F</font> is <font color="Olive" title="b3">V</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) </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="ref" href="measur11.html#T86" target="_self" onmouseover="rs('measur11/T86')" onmouseout="rh()">Th85</a></span>;<br/>
<a NAME="E3:117"/><span class="lab"><font color="Green" title="E83">A2</font></span>: 
 <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> :  ex <font color="Olive" title="b2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b2">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</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="Olive" title="b1">E</font>,<font color="Olive" title="b3">y</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> ) &amp; (  for <font color="Olive" title="b3">V</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <font color="Olive" title="b2">F</font> is <font color="Olive" title="b3">V</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T82" target="_self" onmouseover="rs('measur11/T82')" onmouseout="rh()">Th81</a></span>;<br/>
 <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</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> = 
 <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="srings_3.html#K2" title="SRINGS_3:func.2">DisUnion</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="srings_3.html#T22" onmouseover="rs('srings_3/T22')" onmouseout="rh()">SRINGS_3:22</a></span>
<br/>.= 
 <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="measur11.html#T7" target="_self" onmouseover="rs('measur11/T7')" onmouseout="rh()">Th1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:117"/>
 <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> :  ex <font color="Olive" title="b2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( (  for <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds  <font color="Olive" title="b2">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</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="Olive" title="b1">E</font>,<font color="Olive" title="b3">y</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> ) &amp; (  for <font color="Olive" title="b3">V</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <font color="Olive" title="b2">F</font> is <font color="Olive" title="b3">V</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="#E3:117"><span class="lab"><font color="Green" title="E83">A2</font></span></a>, <a class="ref" href="measur11.html#T88" target="_self" onmouseover="rs('measur11/T88')" onmouseout="rh()">Th87</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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