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<span class="kw">let </span><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S1</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</font><br/>  for <font color="Olive" title="b2">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b3">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S1</font><br/>  for <font color="Olive" title="b4">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b2">S2</font><br/>  for <font color="Olive" title="b5">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>)</span>)</span><br/>  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">S1</font>  st <font color="Olive" title="b5">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>)</span>)</span> holds <br/> ex <font color="Olive" title="b7">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="b8">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  <font color="Olive" title="b7">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b8">x</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="b8">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b6">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b8">P</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="b7">F</font> is <font color="Olive" title="b8">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S1</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b2">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b3">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S2</font><br/>  for <font color="Olive" title="b4">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Olive" title="b1">S2</font>)</span>)</span><br/>  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="c3">S1</font>  st <font color="Olive" title="b4">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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="Olive" title="b1">S2</font>)</span>)</span> holds <br/> 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">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  <font color="Olive" title="b6">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">x</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="b7">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b5">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b7">P</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="b6">F</font> is <font color="Olive" title="b7">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">S2</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b2">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font><br/>  for <font color="Olive" title="b3">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span><br/>  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="c3">S1</font>  st <font color="Olive" title="b3">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> holds <br/> 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">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  <font color="Olive" title="b5">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</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="b6">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b4">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b6">P</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="b5">F</font> is <font color="Olive" title="b6">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">M1</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font><br/>  for <font color="Olive" title="b2">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span><br/>  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="c3">S1</font>  st <font color="Olive" title="b2">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> holds <br/> 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">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  <font color="Olive" title="b4">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">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="Olive" title="b2">E</font>,<font color="Olive" title="b5">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b3">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b5">P</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">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">M2</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span><br/>  for <font color="Olive" title="b2">V</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">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> holds <br/> 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">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  <font color="Olive" title="b3">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">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="Olive" title="b1">E</font>,<font color="Olive" title="b4">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b2">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b4">P</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">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</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="c7">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> holds <br/> 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">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  <font color="Olive" title="b2">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">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="c7">E</font>,<font color="Olive" title="b3">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Olive" title="b1">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b3">P</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">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">V</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="c7">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> implies  ex <font color="Olive" title="b1">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="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  <font color="Olive" title="b1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">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="c7">E</font>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b2">P</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="b1">F</font> is <font color="Olive" title="b2">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) ) )</span><br/>













<span class="kw">assume </span><a NAME="E1:103"/><span class="lab"><font color="Green" title="E71">A1</font></span>: 
<font color="Maroon" title="c7">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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 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="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="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  <font color="Olive" title="b1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">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="c7">E</font>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b2">P</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="b1">F</font> is <font color="Olive" title="b2">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/>

<a NAME="E2:103"/>
<font color="Maroon" title="c2">X2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure1.html#T7" onmouseover="rs('measure1/T7')" onmouseout="rh()">MEASURE1:7</a></span>;<br/>
<a NAME="E3:103"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> : verum </span> } </span> 
 
;<br/>
<a NAME="E4:103"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A2</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur10.html#D5" onmouseover="rs('measur10/D5')" onmouseout="rh()">MEASUR10:def 5</a></span>;<br/>
<a NAME="E5:103"/>
 <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D9" onmouseover="rs('prob_1/D9')" onmouseout="rh()">PROB_1:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">E1</font> = <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:103"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="ref" href="finsub_1.html#D2" onmouseover="rs('finsub_1/D2')" onmouseout="rh()">FINSUB_1:def 2</a></span>;<br/>
<a NAME="E7:103"/><span class="lab"><font color="Green" title="E73">A3</font></span>: 
 <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="srings_3.html#T21" onmouseover="rs('srings_3/T21')" onmouseout="rh()">SRINGS_3:21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1"><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="E1:103_1"/>
( <font color="Maroon" title="c9">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c9">E1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_1"><span class="kw">suppose </span></a><a NAME="E1:103_1_1"/><span class="lab"><font color="Green" title="E74">A4</font></span>: 
<font color="Maroon" title="c9">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="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="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  <font color="Olive" title="b1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">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="c7">E</font>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b2">P</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="b1">F</font> is <font color="Olive" title="b2">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c10">A</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  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#T34" onmouseover="rs('measure1/T34')" onmouseout="rh()">MEASURE1:34</a></span>;<br/><a NAME="E3:103_1_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">F</font> = <font color="Maroon" title="c2">X2</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T45" onmouseover="rs('funcop_1/T45')" onmouseout="rh()">FUNCOP_1:45</a>, <a class="ref" href="numbers.html#T31" onmouseover="rs('numbers/T31')" onmouseout="rh()">NUMBERS:31</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c11">F</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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  <font color="Maroon" title="c11">F</font> <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> <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="c7">E</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">P</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="Maroon" title="c11">F</font> is <font color="Olive" title="b1">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E5:103_1_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> holds  <font color="Maroon" title="c11">F</font> <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> <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="c7">E</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">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">  for <font color="Olive" title="b1">P</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="Maroon" title="c11">F</font> is <font color="Olive" title="b1">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="103_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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="c11">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">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="c7">E</font>,<font color="Maroon" title="c12">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span></span><br/>

<a NAME="E1:103_1_1_1"/><span class="lab"><font color="Green" title="E75">A5</font></span>: 
<font color="Maroon" title="c2">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c2">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D3" onmouseover="rs('subset_1/D3')" onmouseout="rh()">SUBSET_1:def 3</a></span>;<br/>
<span class="p1">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c12">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font> = 
<span class="p1">(<span class="default"><a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c12">x</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<span class="p3">(<span class="default"><a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <font color="Maroon" title="c2">X2</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>,<font color="Maroon" title="c12">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103_1_1_1"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="ref" href="measur11.html#T22" target="_self" onmouseover="rs('measur11/T22')" onmouseout="rh()">Th16</a></span>
<br/>.= 
 <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>,<font color="Maroon" title="c12">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103_1_1"><span class="lab"><font color="Green" title="E74">A4</font></span></a>, <a class="txt" href="#E1:103_1_1_1"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="ref" href="measur11.html#T27" target="_self" onmouseover="rs('measur11/T27')" onmouseout="rh()">Th21</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/>
<a NAME="E3:103_1_1_1"/><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"><span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c12">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D19" onmouseover="rs('valued_0/D19')" onmouseout="rh()">VALUED_0:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:103_1_1_1"/>
<font color="Maroon" title="c11">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">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="c7">E</font>,<font color="Maroon" title="c12">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</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">thus </span><a NAME="E6:103_1_1"/>
 for <font color="Olive" title="b1">P</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="Maroon" title="c11">F</font> is <font color="Olive" title="b1">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="103_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">P</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="c11">F</font> is <font color="Maroon" title="c12">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/>

<a NAME="E1:103_1_1_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> holds  <font color="Maroon" title="c11">F</font> <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#K1" title="MEASUR11:func.1">chi</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c2">X2</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="103_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">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="c11">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E1:103_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:103_1_1_2_1"/>
<font color="Maroon" title="c11">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</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="E2:103_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c10">A</font>,<font color="Maroon" title="c2">X2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D8" onmouseover="rs('funct_2/D8')" onmouseout="rh()">FUNCT_2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:103_1_1_2"/>
<font color="Maroon" title="c11">F</font> is <font color="Maroon" title="c12">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc2.html#T29" onmouseover="rs('mesfunc2/T29')" onmouseout="rh()">MESFUNC2:29</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="103_1_2"><span class="kw">suppose </span></a><a NAME="E1:103_1_2"/><span class="lab"><font color="Green" title="E74">A6</font></span>: 
<font color="Maroon" title="c9">E1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="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="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  <font color="Olive" title="b1">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">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="c7">E</font>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b2">P</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="b1">F</font> is <font color="Olive" title="b2">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/><div class="add"><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</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="c7">E</font>,$1)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">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><span class="kw"> such that </span><br/><a NAME="E3:103_1_2"/><span class="lab"><font color="Green" title="E75">A7</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> holds  <font color="Maroon" title="c10">F</font> <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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span><span class="kw">consider </span><font color="Maroon" title="c11">f</font> being   <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>), <font color="Maroon" title="c12">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c13">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font>, <font color="Maroon" title="c14">Xf</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</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 href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>), <font color="Maroon" title="c15">If</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>), <font color="Maroon" title="c16">Jf</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">X2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)<span class="kw"> such that </span><br/><a NAME="E5:103_1_2"/><span class="lab"><font color="Green" title="E76">A8</font></span>: 
( <font color="Maroon" title="c7">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c11">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Xf</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">If</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">Jf</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">Xf</font> holds <br/><font color="Maroon" title="c14">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<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>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Xf</font></span>)</span> <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>,<span class="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>) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font><br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">If</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">If</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">Xf</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) ) &amp; (  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">P</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">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">If</font> holds <br/><font color="Maroon" title="c15">If</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Olive" title="b2">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <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="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) <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#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c15">If</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">If</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> ) &amp; (  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> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">Jf</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">Jf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c14">Xf</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) ) &amp; (  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">P</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">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">Jf</font> holds <br/><font color="Maroon" title="c16">Jf</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Olive" title="b2">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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   <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) <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#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c16">Jf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">Jf</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:103"><span class="lab"><font color="Green" title="E73">A3</font></span></a>, <a class="txt" href="#E1:103"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="txt" href="#E4:103"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="ref" href="finsub_1.html#D2" onmouseover="rs('finsub_1/D2')" onmouseout="rh()">FINSUB_1:def 2</a>, <a class="txt" href="#E1:103_1_2"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="ref" href="measur11.html#T76" target="_self" onmouseover="rs('measur11/T76')" onmouseout="rh()">Th75</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c10">F</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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  <font color="Maroon" title="c10">F</font> <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> <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="c7">E</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">P</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="Maroon" title="c10">F</font> is <font color="Olive" title="b1">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E6:103_1_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> holds  <font color="Maroon" title="c10">F</font> <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> <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="c7">E</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:103_1_2"><span class="lab"><font color="Green" title="E75">A7</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">  for <font color="Olive" title="b1">P</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="Maroon" title="c10">F</font> is <font color="Olive" title="b1">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><a NAME="E7:103_1_2"/><span class="lab"><font color="Green" title="E77">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">Jf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:103_1_2"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/><a NAME="E8:103_1_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> holds  <font color="Maroon" title="c10">F</font> <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"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c16">Jf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">Jf</font></span>)</span></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="103_1_2_1"><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="c10">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</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#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c16">Jf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">Jf</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c16">Jf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">Jf</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</font> = 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">E</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p4"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>,<span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Maroon" title="c17">x</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:103_1_2"><span class="lab"><font color="Green" title="E76">A8</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"><span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c7">E</font>,<font color="Maroon" title="c17">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <font color="Maroon" title="c8">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T71" target="_self" onmouseover="rs('measur11/T71')" onmouseout="rh()">Th67</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:103_1_2_1"/>
<font color="Maroon" title="c10">F</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c17">x</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#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c16">Jf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">Jf</font></span>)</span></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="#E3:103_1_2"><span class="lab"><font color="Green" title="E75">A7</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="E9:103_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A10</font></span>: 
<font color="Maroon" title="c10">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c16">Jf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">Jf</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D8" onmouseover="rs('funct_2/D8')" onmouseout="rh()">FUNCT_2:def 8</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c17">P</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="c10">F</font> is <font color="Maroon" title="c17">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><a NAME="E10:103_1_2"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">Jf</font> holds <br/><font color="Maroon" title="c16">Jf</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c17">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:103_1_2"><span class="lab"><font color="Green" title="E76">A8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:103_1_2"/>
<font color="Maroon" title="c10">F</font> is <font color="Maroon" title="c17">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:103_1_2"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E7:103_1_2"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="#E9:103_1_2"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="ref" href="measur11.html#T68" target="_self" onmouseover="rs('measur11/T68')" onmouseout="rh()">Th64</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>
