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<span class="kw">let </span><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S1</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</font><br/>  for <font color="Olive" title="b2">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b3">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S1</font><br/>  for <font color="Olive" title="b4">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b2">S2</font><br/>  for <font color="Olive" title="b5">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>)</span>)</span>  st <font color="Olive" title="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> &amp; <font color="Olive" title="b5">E</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>  holds <br/> ex <font color="Olive" title="b6">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="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>) ex <font color="Olive" title="b7">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b1">S1</font> ex <font color="Olive" title="b8">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b2">S2</font> ex <font color="Olive" title="b9">C</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>) ex <font color="Olive" title="b10">I</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>) ex <font color="Olive" title="b11">J</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>) st <br/>( <font color="Olive" title="b5">E</font> <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="Olive" title="b6">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">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="Olive" title="b6">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">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="Olive" title="b7">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">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="Olive" title="b8">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">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="Olive" title="b9">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">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="Olive" title="b10">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">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="Olive" title="b11">J</font> &amp; (  for <font color="Olive" title="b12">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b12">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="Olive" title="b9">C</font> holds <br/><font color="Olive" title="b9">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b12">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="Olive" title="b6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b12">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="Olive" title="b9">C</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="Olive" title="b9">C</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> (<font color="Olive" title="b5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b12">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="b13">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b13">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="Olive" title="b10">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b10">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b13">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b12">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="Olive" title="b4">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="Olive" title="b9">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b13">n</font></span>)</span>,<font color="Olive" title="b12">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b12">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b13">P</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="b12">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="Olive" title="b10">I</font> holds <br/><font color="Olive" title="b10">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b12">n</font> is <font color="Olive" title="b13">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b12">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="Olive" title="b4">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="Olive" title="b9">C</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="Olive" title="b9">C</font></span>)</span></span>)</span>,<font color="Olive" title="b12">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="Olive" title="b10">I</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="Olive" title="b10">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b12">x</font> ) &amp; (  for <font color="Olive" title="b12">y</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="b13">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b13">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="Olive" title="b11">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b11">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b13">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b3">M1</font>,<span class="p1">(<span class="default"><a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p2">(<span class="default"><font color="Olive" title="b9">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b13">n</font></span>)</span>,<font color="Olive" title="b12">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b12">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b13">P</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">S2</font>  st <font color="Olive" title="b12">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="Olive" title="b11">J</font> holds <br/><font color="Olive" title="b11">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b12">n</font> is <font color="Olive" title="b13">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b12">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   <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b3">M1</font>,<span class="p1">(<span class="default"><a href="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="Olive" title="b9">C</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="Olive" title="b9">C</font></span>)</span></span>)</span>,<font color="Olive" title="b12">y</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="Olive" title="b11">J</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="Olive" title="b11">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b12">y</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S1</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b2">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b3">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S2</font><br/>  for <font color="Olive" title="b4">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Olive" title="b1">S2</font>)</span>)</span>  st <font color="Olive" title="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> &amp; <font color="Olive" title="b4">E</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>  holds <br/> ex <font color="Olive" title="b5">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="Olive" title="b1">S2</font>) ex <font color="Olive" title="b6">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b7">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b1">S2</font> ex <font color="Olive" title="b8">C</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>) ex <font color="Olive" title="b9">I</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>) ex <font color="Olive" title="b10">J</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>) st <br/>( <font color="Olive" title="b4">E</font> <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="Olive" title="b5">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">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="Olive" title="b5">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">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="Olive" title="b6">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">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="Olive" title="b7">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">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="Olive" title="b8">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">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="Olive" title="b9">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">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="Olive" title="b10">J</font> &amp; (  for <font color="Olive" title="b11">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b11">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="Olive" title="b8">C</font> holds <br/><font color="Olive" title="b8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">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="Olive" title="b5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">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="Olive" title="b8">C</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="Olive" title="b8">C</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> (<font color="Olive" title="b4">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b11">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="b12">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b12">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="Olive" title="b9">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b9">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b12">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">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="Olive" title="b3">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="Olive" title="b8">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b12">n</font></span>)</span>,<font color="Olive" title="b11">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b11">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b12">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="b11">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="Olive" title="b9">I</font> holds <br/><font color="Olive" title="b9">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b11">n</font> is <font color="Olive" title="b12">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b11">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="Olive" title="b3">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="Olive" title="b8">C</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="Olive" title="b8">C</font></span>)</span></span>)</span>,<font color="Olive" title="b11">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="Olive" title="b9">I</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="Olive" title="b9">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b11">x</font> ) &amp; (  for <font color="Olive" title="b11">y</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="b12">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b12">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="Olive" title="b10">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b10">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b12">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M1</font>,<span class="p1">(<span class="default"><a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p2">(<span class="default"><font color="Olive" title="b8">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b12">n</font></span>)</span>,<font color="Olive" title="b11">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b11">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b12">P</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S2</font>  st <font color="Olive" title="b11">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="Olive" title="b10">J</font> holds <br/><font color="Olive" title="b10">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b11">n</font> is <font color="Olive" title="b12">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b11">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   <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M1</font>,<span class="p1">(<span class="default"><a href="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="Olive" title="b8">C</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="Olive" title="b8">C</font></span>)</span></span>)</span>,<font color="Olive" title="b11">y</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="Olive" title="b10">J</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="Olive" title="b10">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b11">y</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">S2</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b2">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font><br/>  for <font color="Olive" title="b3">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>  st <font color="Olive" title="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> &amp; <font color="Olive" title="b3">E</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>  holds <br/> ex <font color="Olive" title="b4">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>) ex <font color="Olive" title="b5">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b6">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b7">C</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>) ex <font color="Olive" title="b8">I</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>) ex <font color="Olive" title="b9">J</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>) st <br/>( <font color="Olive" title="b3">E</font> <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="Olive" title="b4">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">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="Olive" title="b4">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">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="Olive" title="b5">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">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="Olive" title="b6">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">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="Olive" title="b7">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">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="Olive" title="b8">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">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="Olive" title="b9">J</font> &amp; (  for <font color="Olive" title="b10">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b10">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="Olive" title="b7">C</font> holds <br/><font color="Olive" title="b7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">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="Olive" title="b4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">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="Olive" title="b7">C</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="Olive" title="b7">C</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> (<font color="Olive" title="b3">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b10">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="b11">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b11">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="Olive" title="b8">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b8">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">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="Olive" title="b2">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="Olive" title="b7">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b11">n</font></span>)</span>,<font color="Olive" title="b10">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b10">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b11">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="b10">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="Olive" title="b8">I</font> holds <br/><font color="Olive" title="b8">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b10">n</font> is <font color="Olive" title="b11">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b10">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="Olive" title="b2">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="Olive" title="b7">C</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="Olive" title="b7">C</font></span>)</span></span>)</span>,<font color="Olive" title="b10">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="Olive" title="b8">I</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="Olive" title="b8">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b10">x</font> ) &amp; (  for <font color="Olive" title="b10">y</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="b11">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b11">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="Olive" title="b9">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b9">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M1</font>,<span class="p1">(<span class="default"><a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p2">(<span class="default"><font color="Olive" title="b7">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b11">n</font></span>)</span>,<font color="Olive" title="b10">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b10">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b11">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="b10">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="Olive" title="b9">J</font> holds <br/><font color="Olive" title="b9">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b10">n</font> is <font color="Olive" title="b11">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b10">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   <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M1</font>,<span class="p1">(<span class="default"><a href="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="Olive" title="b7">C</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="Olive" title="b7">C</font></span>)</span></span>)</span>,<font color="Olive" title="b10">y</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="Olive" title="b9">J</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="Olive" title="b9">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b10">y</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">M1</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font><br/>  for <font color="Olive" title="b2">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>  st <font color="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> &amp; <font color="Olive" title="b2">E</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>  holds <br/> ex <font color="Olive" title="b3">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>) ex <font color="Olive" title="b4">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b5">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b6">C</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>) ex <font color="Olive" title="b7">I</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>) ex <font color="Olive" title="b8">J</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>) st <br/>( <font color="Olive" title="b2">E</font> <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="Olive" title="b3">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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="Olive" title="b3">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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="Olive" title="b4">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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="Olive" title="b5">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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="Olive" title="b6">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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="Olive" title="b7">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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="Olive" title="b8">J</font> &amp; (  for <font color="Olive" title="b9">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b9">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="Olive" title="b6">C</font> holds <br/><font color="Olive" title="b6">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">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="Olive" title="b3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">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="Olive" title="b6">C</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="Olive" title="b6">C</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> (<font color="Olive" title="b2">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b9">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="b10">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b10">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="Olive" title="b7">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b7">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">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="Olive" title="b1">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="Olive" title="b6">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b10">n</font></span>)</span>,<font color="Olive" title="b9">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b9">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b10">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="b9">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="Olive" title="b7">I</font> holds <br/><font color="Olive" title="b7">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b9">n</font> is <font color="Olive" title="b10">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b9">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="Olive" title="b1">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="Olive" title="b6">C</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="Olive" title="b6">C</font></span>)</span></span>)</span>,<font color="Olive" title="b9">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="Olive" title="b7">I</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="Olive" title="b7">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b9">x</font> ) &amp; (  for <font color="Olive" title="b9">y</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="b10">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b10">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="Olive" title="b8">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b8">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">y</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="Olive" title="b6">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b10">n</font></span>)</span>,<font color="Olive" title="b9">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b9">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b10">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="b9">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="Olive" title="b8">J</font> holds <br/><font color="Olive" title="b8">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b9">n</font> is <font color="Olive" title="b10">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b9">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   <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="Olive" title="b6">C</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="Olive" title="b6">C</font></span>)</span></span>)</span>,<font color="Olive" title="b9">y</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="Olive" title="b8">J</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="Olive" title="b8">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b9">y</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">M2</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>  st <font color="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> &amp; <font color="Olive" title="b1">E</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>  holds <br/> ex <font color="Olive" title="b2">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>) ex <font color="Olive" title="b3">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b4">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b5">C</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>) ex <font color="Olive" title="b6">I</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>) ex <font color="Olive" title="b7">J</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>) st <br/>( <font color="Olive" title="b1">E</font> <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="Olive" title="b2">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">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="Olive" title="b2">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">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="Olive" title="b3">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">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="Olive" title="b4">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">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="Olive" title="b5">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">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="Olive" title="b6">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">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="Olive" title="b7">J</font> &amp; (  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b8">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="Olive" title="b5">C</font> holds <br/><font color="Olive" title="b5">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">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="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">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="Olive" title="b5">C</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="Olive" title="b5">C</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> (<font color="Olive" title="b1">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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="c1">X1</font><br/>  for <font color="Olive" title="b9">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b9">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="Olive" title="b6">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b6">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">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="Olive" title="b5">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b9">n</font></span>)</span>,<font color="Olive" title="b8">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b9">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="b8">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="Olive" title="b6">I</font> holds <br/><font color="Olive" title="b6">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b8">n</font> is <font color="Olive" title="b9">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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="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="Olive" title="b5">C</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="Olive" title="b5">C</font></span>)</span></span>)</span>,<font color="Olive" title="b8">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="Olive" title="b6">I</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="Olive" title="b6">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b8">x</font> ) &amp; (  for <font color="Olive" title="b8">y</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="b9">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b9">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="Olive" title="b7">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b7">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">y</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="Olive" title="b5">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b9">n</font></span>)</span>,<font color="Olive" title="b8">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b9">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="b8">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="Olive" title="b7">J</font> holds <br/><font color="Olive" title="b7">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b8">n</font> is <font color="Olive" title="b9">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b8">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   <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="Olive" title="b5">C</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="Olive" title="b5">C</font></span>)</span></span>)</span>,<font color="Olive" title="b8">y</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="Olive" title="b7">J</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="Olive" title="b7">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b8">y</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> &amp; <font color="Maroon" title="c7">E</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>  implies  ex <font color="Olive" title="b1">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>) ex <font color="Olive" title="b2">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b3">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b4">C</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>) ex <font color="Olive" title="b5">I</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>) ex <font color="Olive" title="b6">J</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>) st <br/>( <font color="Maroon" title="c7">E</font> <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="Olive" title="b1">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b1">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b2">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b3">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b4">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b5">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b6">J</font> &amp; (  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">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="Olive" title="b4">C</font> holds <br/><font color="Olive" title="b4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">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="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">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="Olive" title="b4">C</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="Olive" title="b4">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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="c1">X1</font><br/>  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b8">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="Olive" title="b5">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b5">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">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="Olive" title="b4">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b8">n</font></span>)</span>,<font color="Olive" title="b7">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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="Maroon" title="c3">S1</font>  st <font color="Olive" title="b7">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="Olive" title="b5">I</font> holds <br/><font color="Olive" title="b5">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b7">n</font> is <font color="Olive" title="b8">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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="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="Olive" title="b4">C</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="Olive" title="b4">C</font></span>)</span></span>)</span>,<font color="Olive" title="b7">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="Olive" title="b5">I</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="Olive" title="b5">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">x</font> ) &amp; (  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><br/>  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b8">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="Olive" title="b6">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b6">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">y</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="Olive" title="b4">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b8">n</font></span>)</span>,<font color="Olive" title="b7">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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="Maroon" title="c4">S2</font>  st <font color="Olive" title="b7">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="Olive" title="b6">J</font> holds <br/><font color="Olive" title="b6">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b7">n</font> is <font color="Olive" title="b8">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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   <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="Olive" title="b4">C</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="Olive" title="b4">C</font></span>)</span></span>)</span>,<font color="Olive" title="b7">y</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="Olive" title="b6">J</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="Olive" title="b6">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">y</font> ) ) )</span><br/>











<span class="kw">assume </span><a NAME="E1:99"/><span class="lab"><font color="Green" title="E69">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> &amp; <font color="Maroon" title="c7">E</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="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>) ex <font color="Olive" title="b2">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b3">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b4">C</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>) ex <font color="Olive" title="b5">I</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>) ex <font color="Olive" title="b6">J</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>) st <br/>( <font color="Maroon" title="c7">E</font> <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="Olive" title="b1">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b1">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b2">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b3">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b4">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b5">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b6">J</font> &amp; (  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">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="Olive" title="b4">C</font> holds <br/><font color="Olive" title="b4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">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="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">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="Olive" title="b4">C</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="Olive" title="b4">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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="c1">X1</font><br/>  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b8">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="Olive" title="b5">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b5">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">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="Olive" title="b4">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b8">n</font></span>)</span>,<font color="Olive" title="b7">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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="Maroon" title="c3">S1</font>  st <font color="Olive" title="b7">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="Olive" title="b5">I</font> holds <br/><font color="Olive" title="b5">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b7">n</font> is <font color="Olive" title="b8">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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="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="Olive" title="b4">C</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="Olive" title="b4">C</font></span>)</span></span>)</span>,<font color="Olive" title="b7">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="Olive" title="b5">I</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="Olive" title="b5">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">x</font> ) &amp; (  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><br/>  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b8">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="Olive" title="b6">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b6">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">y</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="Olive" title="b4">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b8">n</font></span>)</span>,<font color="Olive" title="b7">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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="Maroon" title="c4">S2</font>  st <font color="Olive" title="b7">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="Olive" title="b6">J</font> holds <br/><font color="Olive" title="b6">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b7">n</font> is <font color="Olive" title="b8">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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   <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="Olive" title="b4">C</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="Olive" title="b4">C</font></span>)</span></span>)</span>,<font color="Olive" title="b7">y</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="Olive" title="b6">J</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="Olive" title="b6">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">y</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">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="c9">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="c10">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="c11">C</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>)<span class="kw"> such that </span><br/><a NAME="E3:99"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
( <font color="Maroon" title="c7">E</font> <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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c9">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c10">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c11">C</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="c8">F</font> holds <br/><font color="Maroon" title="c8">F</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> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &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="c11">C</font> holds <br/><font color="Maroon" title="c11">C</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="c8">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="c11">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &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">x</font>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c11">C</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><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="c10">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font></span>)</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> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c7">E</font>,<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="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>) ) &amp; (  for <font color="Olive" title="b1">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   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c7">E</font>,<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="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="measur11.html#T74" target="_self" onmouseover="rs('measur11/T74')" onmouseout="rh()">Th70</a></span>;<br/>
<a NAME="E4:99"/><span class="lab"><font color="Green" title="E71">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="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">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> 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="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds  <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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> $1</span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span>) ) );<br/>
<a NAME="E5:99"/><span class="lab"><font color="Green" title="E72">I1</font></span>: 
 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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">z</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> implies  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">n</font>,<font color="Olive" title="b1">z</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_1"/>
<font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">n</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</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>  =  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>,$1)</span>)</span>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E3:99_1"/><span class="lab"><font color="Green" title="E73">I2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds  <font color="Maroon" title="c13">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">take </span><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c13">f</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c14">z</font>]</span><br/>

<span class="kw">thus </span><a NAME="E4:99_1"/>
 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="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">z</font> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds  <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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_1"><span class="lab"><font color="Green" title="E73">I2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c12">I</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E7:99"/><span class="lab"><font color="Green" title="E73">I3</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> &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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c12">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" onmouseover="rs('finseq_1/S1')" onmouseout="rh()">FINSEQ_1:sch 1</a>(<a class="txt" href="#E5:99"><span class="lab"><font color="Green" title="E72">I1</font></span></a>);<br/></span>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c12">I</font> holds <br/><font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">z</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c12">I</font> implies <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>) )</span><br/><span class="kw">assume </span><a NAME="E1:99_2"/>
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c12">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:99_2"/><span class="lab"><font color="Green" title="E74">I4</font></span>: 
( <font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">I</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">n</font> = <font color="Maroon" title="c14">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2"><span class="lab"><font color="Green" title="E74">I4</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c16">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E6:99_2"/><span class="lab"><font color="Green" title="E75">I5</font></span>: 
( <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</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="c1">X1</font> holds  <font color="Maroon" title="c16">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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c15">n</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="txt" href="#E3:99_2"><span class="lab"><font color="Green" title="E74">I4</font></span></a></span>;<br/><a NAME="E7:99_2"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X1</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:99_2"/>
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_2"><span class="lab"><font color="Green" title="E74">I4</font></span></a>, <a class="txt" href="#E6:99_2"><span class="lab"><font color="Green" title="E75">I5</font></span></a>, <a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:99"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c12">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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>)
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">I</font> = <font color="Maroon" title="c12">I</font> as    <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>) <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E11:99"/><span class="lab"><font color="Green" title="E74">I6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font><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="c13">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">I</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="c11">C</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>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c13">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c14">x</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c15">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">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="c13">I</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c15">n</font></span>)</span>,<font color="Maroon" title="c14">x</font>)</span>)</span>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:99_3"/><span class="lab"><font color="Green" title="E75">I7</font></span>: 
<font color="Maroon" title="c15">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="c13">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c15">n</font></span>)</span>,<font color="Maroon" title="c14">x</font>)</span>)</span>)</span><br/>

<a NAME="E2:99_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">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="c8">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">Fn</font> = <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/>
<a NAME="E4:99_3"/>
 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="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds  <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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c15">n</font></span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span>) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="txt" href="#E1:99_3"><span class="lab"><font color="Green" title="E75">I7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:99_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c15">n</font></span>)</span>,<font color="Maroon" title="c14">x</font>)</span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E75">I7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> 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="c13">I</font> holds <br/><font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">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="c13">I</font> implies <font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:99_4"/><span class="lab"><font color="Green" title="E76">I8</font></span>: 
<font color="Maroon" title="c14">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="c13">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E3:99_4"/><span class="lab"><font color="Green" title="E77">I9</font></span>: 
( <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</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="c1">X1</font> holds  <font color="Maroon" title="c15">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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a></span>;<br/><a NAME="E4:99_4"/><span class="lab"><font color="Green" title="E78">I10</font></span>: 
<font color="Maroon" title="c14">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="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="txt" href="#E1:99_4"><span class="lab"><font color="Green" title="E76">I8</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">Fn</font> = <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/><a NAME="E6:99_4"/>
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_4"><span class="lab"><font color="Green" title="E78">I10</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/><a NAME="E7:99_4"/><span class="kw">then </span>
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measur10.html#D5" onmouseover="rs('measur10/D5')" onmouseout="rh()">MEASUR10:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">An</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c18">Bn</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E9:99_4"/><span class="lab"><font color="Green" title="E79">I11</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c17">An</font>,<font color="Maroon" title="c18">Bn</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 ;<br/><a NAME="E10:99_4"/>
 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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">f</font> <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="99_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font></span><br/>

<a NAME="E1:99_4_1"/><span class="lab"><font color="Green" title="E80">I12</font></span>: 
<font color="Maroon" title="c14">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="c11">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="txt" href="#E1:99_4"><span class="lab"><font color="Green" title="E76">I8</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E2:99_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E3:99_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">I13</font></span>: 
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c14">n</font> <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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E1:99_4_1"><span class="lab"><font color="Green" title="E80">I12</font></span></a></span>;<br/>
<a NAME="E4:99_4_1"/>
<font color="Maroon" title="c15">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c14">n</font></span>)</span>,<font color="Maroon" title="c19">x</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_4"><span class="lab"><font color="Green" title="E77">I9</font></span></a></span>;<br/>
<a NAME="E5:99_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">I14</font></span>: 
<font color="Maroon" title="c15">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c16">Fn</font>,<font color="Maroon" title="c19">x</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c17">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_4"><span class="lab"><font color="Green" title="E79">I11</font></span></a>, <a class="txt" href="#E3:99_4_1"><span class="lab"><font color="Green" title="E81">I13</font></span></a>, <a class="ref" href="measur11.html#T69" target="_self" onmouseover="rs('measur11/T69')" onmouseout="rh()">Th65</a></span>;<br/>
<a NAME="E6:99_4_1"/>
( <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c16">Fn</font>,<font color="Maroon" title="c19">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c17">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T51" onmouseover="rs('supinf_2/T51')" onmouseout="rh()">SUPINF_2:51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:99_4_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c19">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99_4_1"><span class="lab"><font color="Green" title="E82">I14</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="E11:99_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">V105</a>() <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_4"><span class="lab"><font color="Green" title="E77">I9</font></span></a>, <a class="ref" href="supinf_2.html#T39" onmouseover="rs('supinf_2/T39')" onmouseout="rh()">SUPINF_2:39</a></span>;<br/><span class="kw">hence </span><a NAME="E12:99_4"/>
<font color="Maroon" title="c13">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E13:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">I15</font></span>: 
<font color="Maroon" title="c13">I</font> is  <a href="measur11.html#V2" title="MEASUR11:attr.2">without_-infty-valued</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">I</font> = <font color="Maroon" title="c13">I</font> as   <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>) ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> 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/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; (  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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> $1</span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span>) ) );<br/>
<a NAME="E15:99"/><span class="lab"><font color="Green" title="E77">J1</font></span>: 
 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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">z</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> implies  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Olive" title="b1">z</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_5"/>
<font color="Maroon" title="c15">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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>  =  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c15">n</font></span>)</span>,$1)</span>)</span>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">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="E3:99_5"/><span class="lab"><font color="Green" title="E78">J2</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="c16">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">take </span><font color="Maroon" title="c17">z</font> = <font color="Maroon" title="c16">f</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c17">z</font>]</span><br/>

<span class="kw">thus </span><a NAME="E4:99_5"/>
 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/>( <font color="Olive" title="b1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c17">z</font> &amp; (  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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c15">n</font></span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_5"><span class="lab"><font color="Green" title="E78">J2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c15">J</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E17:99"/><span class="lab"><font color="Green" title="E78">J3</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> &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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c15">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" onmouseover="rs('finseq_1/S1')" onmouseout="rh()">FINSEQ_1:sch 1</a>(<a class="txt" href="#E15:99"><span class="lab"><font color="Green" title="E77">J1</font></span></a>);<br/></span>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_6"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c15">J</font> holds <br/><font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">z</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c15">J</font> implies <font color="Maroon" title="c16">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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><br/><span class="kw">assume </span><a NAME="E1:99_6"/>
<font color="Maroon" title="c16">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c15">J</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:99_6"/><span class="lab"><font color="Green" title="E79">J4</font></span>: 
( <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">J</font> &amp; <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">n</font> = <font color="Maroon" title="c17">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_6"><span class="lab"><font color="Green" title="E79">J4</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">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="E6:99_6"/><span class="lab"><font color="Green" title="E80">J5</font></span>: 
( <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</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> holds  <font color="Maroon" title="c19">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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="txt" href="#E3:99_6"><span class="lab"><font color="Green" title="E79">J4</font></span></a></span>;<br/><a NAME="E7:99_6"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X2</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:99_6"/>
<font color="Maroon" title="c16">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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">by</span> <span class="lab"><a class="txt" href="#E3:99_6"><span class="lab"><font color="Green" title="E79">J4</font></span></a>, <a class="txt" href="#E6:99_6"><span class="lab"><font color="Green" title="E80">J5</font></span></a>, <a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E19:99"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c15">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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>)
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">J</font> = <font color="Maroon" title="c15">J</font> as    <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">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E21:99"/><span class="lab"><font color="Green" title="E79">J6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b2">n</font> 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">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">J</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="c11">C</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>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c17">x</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c18">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">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">J</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Maroon" title="c17">x</font>)</span>)</span>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:99_7"/><span class="lab"><font color="Green" title="E80">J7</font></span>: 
<font color="Maroon" title="c18">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">J</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Maroon" title="c17">x</font>)</span>)</span>)</span><br/>

<a NAME="E2:99_7"/><span class="kw">then </span>
<font color="Maroon" title="c18">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="c8">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">Fn</font> = <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/>
<a NAME="E4:99_7"/>
 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/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> &amp; (  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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span>) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="txt" href="#E1:99_7"><span class="lab"><font color="Green" title="E80">J7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:99_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Maroon" title="c17">x</font>)</span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E80">J7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_8"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> 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">J</font> holds <br/><font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">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">J</font> implies <font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:99_8"/><span class="lab"><font color="Green" title="E81">J8</font></span>: 
<font color="Maroon" title="c17">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">J</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">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="E3:99_8"/><span class="lab"><font color="Green" title="E82">J9</font></span>: 
( <font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</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> holds  <font color="Maroon" title="c18">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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c17">n</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a></span>;<br/><a NAME="E4:99_8"/><span class="lab"><font color="Green" title="E83">J10</font></span>: 
<font color="Maroon" title="c17">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="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="txt" href="#E1:99_8"><span class="lab"><font color="Green" title="E81">J8</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">Fn</font> = <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/><a NAME="E6:99_8"/>
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_8"><span class="lab"><font color="Green" title="E83">J10</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/><a NAME="E7:99_8"/><span class="kw">then </span>
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measur10.html#D5" onmouseover="rs('measur10/D5')" onmouseout="rh()">MEASUR10:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">An</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c21">Bn</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E9:99_8"/><span class="lab"><font color="Green" title="E84">J11</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c20">An</font>,<font color="Maroon" title="c21">Bn</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 ;<br/><a NAME="E10:99_8"/>
 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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">f</font> <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="99_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">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">  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">x</font></span><br/>

<a NAME="E1:99_8_1"/><span class="lab"><font color="Green" title="E85">J12</font></span>: 
<font color="Maroon" title="c17">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="c11">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="txt" href="#E1:99_8"><span class="lab"><font color="Green" title="E81">J8</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E2:99_8_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E3:99_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">J13</font></span>: 
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c17">n</font> <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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E1:99_8_1"><span class="lab"><font color="Green" title="E85">J12</font></span></a></span>;<br/>
<a NAME="E4:99_8_1"/>
<font color="Maroon" title="c18">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c17">n</font></span>)</span>,<font color="Maroon" title="c22">x</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_8"><span class="lab"><font color="Green" title="E82">J9</font></span></a></span>;<br/>
<a NAME="E5:99_8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">J14</font></span>: 
<font color="Maroon" title="c18">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c19">Fn</font>,<font color="Maroon" title="c22">x</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">Bn</font>,<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="c22">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_8"><span class="lab"><font color="Green" title="E84">J11</font></span></a>, <a class="txt" href="#E3:99_8_1"><span class="lab"><font color="Green" title="E86">J13</font></span></a>, <a class="ref" href="measur11.html#T69" target="_self" onmouseover="rs('measur11/T69')" onmouseout="rh()">Th65</a></span>;<br/>
<a NAME="E6:99_8_1"/>
( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c19">Fn</font>,<font color="Maroon" title="c22">x</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">Bn</font>,<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="c22">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T51" onmouseover="rs('supinf_2/T51')" onmouseout="rh()">SUPINF_2:51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:99_8_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99_8_1"><span class="lab"><font color="Green" title="E87">J14</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="E11:99_8"/><span class="kw">then </span>
<font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> is  <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">V105</a>() <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="txt" href="#E3:99_8"><span class="lab"><font color="Green" title="E82">J9</font></span></a>, <a class="ref" href="supinf_2.html#T39" onmouseover="rs('supinf_2/T39')" onmouseout="rh()">SUPINF_2:39</a></span>;<br/><span class="kw">hence </span><a NAME="E12:99_8"/>
<font color="Maroon" title="c16">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E23:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">J15</font></span>: 
<font color="Maroon" title="c16">J</font> is  <a href="measur11.html#V2" title="MEASUR11:attr.2">without_-infty-valued</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">J</font> = <font color="Maroon" title="c16">J</font> as   <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>) ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">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">  ex <font color="Olive" title="b1">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b2">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b3">C</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>) ex <font color="Olive" title="b4">I</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>) ex <font color="Olive" title="b5">J</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>) st <br/>( <font color="Maroon" title="c7">E</font> <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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b1">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b2">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b3">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b4">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b5">J</font> &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b6">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="Olive" title="b3">C</font> holds <br/><font color="Olive" title="b3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">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="Olive" title="b3">C</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="Olive" title="b3">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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="c1">X1</font><br/>  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">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="Olive" title="b4">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">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="Olive" title="b3">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b7">n</font></span>)</span>,<font color="Olive" title="b6">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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="Maroon" title="c3">S1</font>  st <font color="Olive" title="b6">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="Olive" title="b4">I</font> holds <br/><font color="Olive" title="b4">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b6">n</font> is <font color="Olive" title="b7">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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="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="Olive" title="b3">C</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="Olive" title="b3">C</font></span>)</span></span>)</span>,<font color="Olive" title="b6">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="Olive" title="b4">I</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="Olive" title="b4">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font> ) &amp; (  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><br/>  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">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="Olive" title="b5">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b5">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">y</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="Olive" title="b3">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b7">n</font></span>)</span>,<font color="Olive" title="b6">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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="Maroon" title="c4">S2</font>  st <font color="Olive" title="b6">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="Olive" title="b5">J</font> holds <br/><font color="Olive" title="b5">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b6">n</font> is <font color="Olive" title="b7">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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   <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="Olive" title="b3">C</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="Olive" title="b3">C</font></span>)</span></span>)</span>,<font color="Olive" title="b6">y</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="Olive" title="b5">J</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="Olive" title="b5">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">y</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">A</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">  ex <font color="Olive" title="b1">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b2">C</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>) ex <font color="Olive" title="b3">I</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>) ex <font color="Olive" title="b4">J</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>) st <br/>( <font color="Maroon" title="c7">E</font> <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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c9">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b1">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b2">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b3">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b4">J</font> &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">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="Olive" title="b2">C</font> holds <br/><font color="Olive" title="b2">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">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="Olive" title="b2">C</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="Olive" title="b2">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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="c1">X1</font><br/>  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b6">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="Olive" title="b3">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">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="Olive" title="b2">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b6">n</font></span>)</span>,<font color="Olive" title="b5">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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="c3">S1</font>  st <font color="Olive" title="b5">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="Olive" title="b3">I</font> holds <br/><font color="Olive" title="b3">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b5">n</font> is <font color="Olive" title="b6">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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="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="Olive" title="b2">C</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="Olive" title="b2">C</font></span>)</span></span>)</span>,<font color="Olive" title="b5">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="Olive" title="b3">I</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="Olive" title="b3">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font> ) &amp; (  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><br/>  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b6">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="Olive" title="b4">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</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="Olive" title="b2">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b6">n</font></span>)</span>,<font color="Olive" title="b5">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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>  st <font color="Olive" title="b5">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="Olive" title="b4">J</font> holds <br/><font color="Olive" title="b4">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b5">n</font> is <font color="Olive" title="b6">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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   <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="Olive" title="b2">C</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="Olive" title="b2">C</font></span>)</span></span>)</span>,<font color="Olive" title="b5">y</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="Olive" title="b4">J</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="Olive" title="b4">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">y</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">B</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">  ex <font color="Olive" title="b1">C</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>) ex <font color="Olive" title="b2">I</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>) ex <font color="Olive" title="b3">J</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>) st <br/>( <font color="Maroon" title="c7">E</font> <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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c9">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c10">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b1">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b2">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b3">J</font> &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">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="Olive" title="b1">C</font> holds <br/><font color="Olive" title="b1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">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="Olive" title="b1">C</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="Olive" title="b1">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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="c1">X1</font><br/>  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">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="Olive" title="b2">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">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="Olive" title="b1">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Olive" title="b4">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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="c3">S1</font>  st <font color="Olive" title="b4">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="Olive" title="b2">I</font> holds <br/><font color="Olive" title="b2">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b4">n</font> is <font color="Olive" title="b5">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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="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="Olive" title="b1">C</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="Olive" title="b1">C</font></span>)</span></span>)</span>,<font color="Olive" title="b4">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="Olive" title="b2">I</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="Olive" title="b2">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font> ) &amp; (  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><br/>  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">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="Olive" title="b3">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">y</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="Olive" title="b1">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Olive" title="b4">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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>  st <font color="Olive" title="b4">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="Olive" title="b3">J</font> holds <br/><font color="Olive" title="b3">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b4">n</font> is <font color="Olive" title="b5">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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   <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="Olive" title="b1">C</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="Olive" title="b1">C</font></span>)</span></span>)</span>,<font color="Olive" title="b4">y</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="Olive" title="b3">J</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="Olive" title="b3">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">y</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">C</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">  ex <font color="Olive" title="b1">I</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>) ex <font color="Olive" title="b2">J</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>) st <br/>( <font color="Maroon" title="c7">E</font> <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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c9">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c10">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c11">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b1">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b2">J</font> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">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="c11">C</font> holds <br/><font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">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="c11">C</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="c11">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font><br/>  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">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="Olive" title="b1">I</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b4">n</font></span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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="c3">S1</font>  st <font color="Olive" title="b3">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="Olive" title="b1">I</font> holds <br/><font color="Olive" title="b1">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b3">n</font> is <font color="Olive" title="b4">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> 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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b3">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="Olive" title="b1">I</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="Olive" title="b1">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font> ) &amp; (  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><br/>  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">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="Olive" title="b2">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b4">n</font></span>)</span>,<font color="Olive" title="b3">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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>  st <font color="Olive" title="b3">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="Olive" title="b2">J</font> holds <br/><font color="Olive" title="b2">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b3">n</font> is <font color="Olive" title="b4">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b3">y</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="Olive" title="b2">J</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="Olive" title="b2">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c14">I</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">  ex <font color="Olive" title="b1">J</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>) st <br/>( <font color="Maroon" title="c7">E</font> <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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c9">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c10">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c11">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="Olive" title="b1">J</font> &amp; (  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="c11">C</font> holds <br/><font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">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="c11">C</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="c11">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font><br/>  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">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">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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="c3">S1</font>  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="c14">I</font> holds <br/><font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b2">n</font> is <font color="Olive" title="b3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b2">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="c14">I</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="c14">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font> ) &amp; (  for <font color="Olive" title="b2">y</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="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">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="Olive" title="b1">J</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Olive" title="b2">y</font>)</span>)</span>) ) &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  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>  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="Olive" title="b1">J</font> holds <br/><font color="Olive" title="b1">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Olive" title="b2">n</font> is <font color="Olive" title="b3">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  ) &amp; (  for <font color="Olive" title="b2">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b2">y</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="Olive" title="b1">J</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="Olive" title="b1">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">y</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c17">J</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="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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c9">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c10">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c11">C</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c17">J</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="c11">C</font> holds <br/><font color="Maroon" title="c11">C</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="c8">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="c11">C</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="c11">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &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="c14">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">I</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="c11">C</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="c14">I</font> holds <br/><font color="Maroon" title="c14">I</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="c11">C</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="c11">C</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="c14">I</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="c14">I</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">y</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="c17">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">J</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">y</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="c11">C</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">y</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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</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">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E25:99"/>
( <font color="Maroon" title="c7">E</font> <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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c9">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c10">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c11">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c17">J</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="c11">C</font> holds <br/><font color="Maroon" title="c11">C</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="c8">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="c11">C</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="c11">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &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="c14">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">I</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="c11">C</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="c14">I</font> holds <br/><font color="Maroon" title="c14">I</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="c11">C</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="c11">C</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="c14">I</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="c14">I</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">y</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="c17">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">J</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">y</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="c11">C</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">y</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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</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">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E26:99"/><span class="lab"><font color="Green" title="E82">K1</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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">I</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c17">J</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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="c11">C</font> holds <br/><font color="Maroon" title="c11">C</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="c8">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="c11">C</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="c11">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &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="c14">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">I</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="c11">C</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="c14">I</font> holds <br/><font color="Maroon" title="c14">I</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="c11">C</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="c11">C</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="c14">I</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="c14">I</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">y</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="c17">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">J</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">y</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="c11">C</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">y</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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</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">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E27:99"/>
 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="c11">C</font> holds <br/><font color="Maroon" title="c11">C</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="c8">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</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"> ( <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">C</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="c11">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &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="c14">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">I</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="c11">C</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="c14">I</font> holds <br/><font color="Maroon" title="c14">I</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="c11">C</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="c11">C</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="c14">I</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="c14">I</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">y</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="c17">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">J</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">y</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="c11">C</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">y</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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</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">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<a NAME="E28:99"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/>
<a NAME="E29:99"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:99"/>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">C</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="c11">C</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> (<font color="Maroon" title="c7">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</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">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="c14">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">I</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="c11">C</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="c14">I</font> holds <br/><font color="Maroon" title="c14">I</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="c11">C</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="c11">C</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="c14">I</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="c14">I</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">y</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="c17">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">J</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">y</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="c11">C</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">y</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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</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">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E31:99"/>
 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="c14">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">I</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="c11">C</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99"><span class="lab"><font color="Green" title="E74">I6</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">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="c14">I</font> holds <br/><font color="Maroon" title="c14">I</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="c11">C</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="c11">C</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="c14">I</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="c14">I</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">y</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="c17">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">J</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">y</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="c11">C</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">y</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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</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">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E32:99"/>
 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="c14">I</font> holds <br/><font color="Maroon" title="c14">I</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> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <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="c11">C</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="c11">C</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="c14">I</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="c14">I</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">y</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="c17">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">J</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">y</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="c11">C</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">y</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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</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">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">y</font> ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="Maroon" title="c18">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">I</font> holds <br/><font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</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">let </span><font color="Maroon" title="c19">P</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="c18">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">I</font> implies <font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:99_9"/><span class="lab"><font color="Green" title="E83">I16</font></span>: 
<font color="Maroon" title="c18">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">I</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="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E3:99_9"/><span class="lab"><font color="Green" title="E84">I17</font></span>: 
( <font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</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="c1">X1</font> holds  <font color="Maroon" title="c20">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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a></span>;<br/>
<a NAME="E4:99_9"/><span class="lab"><font color="Green" title="E85">I18</font></span>: 
<font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c20">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_9"><span class="lab"><font color="Green" title="E83">I16</font></span></a>, <a class="txt" href="#E3:99_9"><span class="lab"><font color="Green" title="E84">I17</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E5:99_9"/><span class="lab"><font color="Green" title="E86">I19</font></span>: 
<font color="Maroon" title="c18">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="c8">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="txt" href="#E1:99_9"><span class="lab"><font color="Green" title="E83">I16</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:99_9"/><span class="kw">then </span>
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/>
<a NAME="E7:99_9"/><span class="kw">then </span>
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur10.html#D5" onmouseover="rs('measur10/D5')" onmouseout="rh()">MEASUR10:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">An</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c22">Bn</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E9:99_9"/><span class="lab"><font color="Green" title="E87">I20</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c22">Bn</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">Fn</font> = <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E5:99_9"><span class="lab"><font color="Green" title="E86">I19</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_9_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="E1:99_9_1"/>
( <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">Bn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">Bn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_9_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_9_1_1"/><span class="lab"><font color="Green" title="E88">I21</font></span>: 
<font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">Bn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><div class="add"><a NAME="E2:99_9_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="c1">X1</font> holds  <font color="Maroon" title="c20">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="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</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="99_9_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font></span><br/>

<a NAME="E1:99_9_1_1_1"/><span class="lab"><font color="Green" title="E89">I22</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Maroon" title="c24">x</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_9"><span class="lab"><font color="Green" title="E84">I17</font></span></a></span>;<br/>
<a NAME="E2:99_9_1_1_1"/><span class="lab"><font color="Green" title="E90">I23</font></span>: 
<font color="Maroon" title="c18">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="c11">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="txt" href="#E1:99_9"><span class="lab"><font color="Green" title="E83">I16</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:99_9_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E4:99_9_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> <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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E2:99_9_1_1_1"><span class="lab"><font color="Green" title="E90">I23</font></span></a></span>;<br/>
<a NAME="E5:99_9_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">I24</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c23">Fn</font>,<font color="Maroon" title="c24">x</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_9"><span class="lab"><font color="Green" title="E87">I20</font></span></a>, <a class="txt" href="#E1:99_9_1_1_1"><span class="lab"><font color="Green" title="E89">I22</font></span></a>, <a class="ref" href="measur11.html#T69" target="_self" onmouseover="rs('measur11/T69')" onmouseout="rh()">Th65</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_9_1_1_1_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="E1:99_9_1_1_1_1"/>
( <font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">An</font> or  not <font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">An</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_9_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_9_1_1_1_1_1"/><span class="lab"><font color="Green" title="E92">I25</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">An</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font></span><br/><div class="add"><a NAME="E2:99_9_1_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c23">Fn</font>,<font color="Maroon" title="c24">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_9"><span class="lab"><font color="Green" title="E87">I20</font></span></a>, <a class="txt" href="#E1:99_9_1_1"><span class="lab"><font color="Green" title="E88">I21</font></span></a>, <a class="ref" href="measur11.html#T22" target="_self" onmouseover="rs('measur11/T22')" onmouseout="rh()">Th16</a>, <a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/><a NAME="E3:99_9_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99_9_1_1_1"><span class="lab"><font color="Green" title="E91">I24</font></span></a>, <a class="ref" href="xxreal_3.html#T81" onmouseover="rs('xxreal_3/T81')" onmouseout="rh()">XXREAL_3:81</a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_9_1_1_1_1_1"/>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_9_1_1_1_1_1"><span class="lab"><font color="Green" title="E92">I25</font></span></a>, <a class="ref" href="measur10.html#D7" onmouseover="rs('measur10/D7')" onmouseout="rh()">MEASUR10:def 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_9_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_9_1_1_1_1_2"/><span class="lab"><font color="Green" title="E92">I26</font></span>: 
 not <font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">An</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font></span><br/><div class="add"><a NAME="E2:99_9_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">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/><a NAME="E3:99_9_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">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="txt" href="#E5:99_9_1_1_1"><span class="lab"><font color="Green" title="E91">I24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_9_1_1_1_1_2"/>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_9_1_1_1_1_2"><span class="lab"><font color="Green" title="E92">I26</font></span></a>, <a class="ref" href="measur10.html#D7" onmouseover="rs('measur10/D7')" onmouseout="rh()">MEASUR10:def 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E3:99_9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</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="E4:99_9_1_1"/>
<font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">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="#E4:99_9"><span class="lab"><font color="Green" title="E85">I18</font></span></a>, <a class="ref" href="measur10.html#T32" onmouseover="rs('measur10/T32')" onmouseout="rh()">MEASUR10:32</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_9_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_9_1_2"/><span class="lab"><font color="Green" title="E88">I27</font></span>: 
<font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">Bn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><div class="add"><a NAME="E2:99_9_1_2"/>
<font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">Bn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T51" onmouseover="rs('supinf_2/T51')" onmouseout="rh()">SUPINF_2:51</a></span>;<br/><a NAME="E3:99_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">Bn</font> <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="txt" href="#E1:99_9_1_2"><span class="lab"><font color="Green" title="E88">I27</font></span></a>, <a class="ref" href="xxreal_0.html#T14" onmouseover="rs('xxreal_0/T14')" onmouseout="rh()">XXREAL_0:14</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">r</font> = <font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c22">Bn</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><a NAME="E5:99_9_1_2"/><span class="lab"><font color="Green" title="E89">I28</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:99_9_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">I29</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X1</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D6" onmouseover="rs('mesfunc1/D6')" onmouseout="rh()">MESFUNC1:def 6</a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E7:99_9_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="c1">X1</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font> holds <br/><font color="Maroon" title="c20">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"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</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="99_9_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font> implies <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_9_1_2_1"/>
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font></span><br/>

<a NAME="E2:99_9_1_2_1"/><span class="lab"><font color="Green" title="E91">I30</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Maroon" title="c25">x</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_9"><span class="lab"><font color="Green" title="E84">I17</font></span></a></span>;<br/>
<a NAME="E3:99_9_1_2_1"/><span class="lab"><font color="Green" title="E92">I31</font></span>: 
<font color="Maroon" title="c18">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="c11">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="txt" href="#E1:99_9"><span class="lab"><font color="Green" title="E83">I16</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:99_9_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E5:99_9_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> <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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_9_1_2_1"><span class="lab"><font color="Green" title="E92">I31</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a></span>;<br/>
<a NAME="E6:99_9_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">I32</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c23">Fn</font>,<font color="Maroon" title="c25">x</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_9"><span class="lab"><font color="Green" title="E87">I20</font></span></a>, <a class="txt" href="#E2:99_9_1_2_1"><span class="lab"><font color="Green" title="E91">I30</font></span></a>, <a class="ref" href="measur11.html#T69" target="_self" onmouseover="rs('measur11/T69')" onmouseout="rh()">Th65</a></span>;<br/>
<a NAME="E7:99_9_1_2_1"/><span class="lab"><font color="Green" title="E94">I33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_9_1_2"><span class="lab"><font color="Green" title="E90">I29</font></span></a>, <a class="ref" href="mesfunc1.html#D6" onmouseover="rs('mesfunc1/D6')" onmouseout="rh()">MESFUNC1:def 6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_9_1_2_1_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="E1:99_9_1_2_1_1"/>
( <font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">An</font> or  not <font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">An</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_9_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_9_1_2_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">An</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:99_9_1_2_1_1_1"/>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99_9_1_2_1"><span class="lab"><font color="Green" title="E94">I33</font></span></a>, <a class="txt" href="#E6:99_9_1_2_1"><span class="lab"><font color="Green" title="E93">I32</font></span></a>, <a class="txt" href="#E9:99_9"><span class="lab"><font color="Green" title="E87">I20</font></span></a>, <a class="ref" href="measur11.html#T22" target="_self" onmouseover="rs('measur11/T22')" onmouseout="rh()">Th16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_9_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_9_1_2_1_1_2"/>
 not <font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c21">An</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font></span><br/><div class="add"><a NAME="E2:99_9_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">I34</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">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/><a NAME="E3:99_9_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_9_1_2"><span class="lab"><font color="Green" title="E90">I29</font></span></a>, <a class="ref" href="mesfunc1.html#D6" onmouseover="rs('mesfunc1/D6')" onmouseout="rh()">MESFUNC1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_9_1_2_1_1_2"/>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_9_1_2_1_1_2"><span class="lab"><font color="Green" title="E95">I34</font></span></a>, <a class="txt" href="#E6:99_9_1_2_1"><span class="lab"><font color="Green" title="E93">I32</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E8:99_9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c1">X1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_9_1_2"><span class="lab"><font color="Green" title="E90">I29</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E9:99_9_1_2"/>
<font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">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="#E4:99_9"><span class="lab"><font color="Green" title="E85">I18</font></span></a>, <a class="txt" href="#E5:99_9_1_2"><span class="lab"><font color="Green" title="E89">I28</font></span></a>, <a class="ref" href="mesfunc2.html#T29" onmouseover="rs('mesfunc2/T29')" onmouseout="rh()">MESFUNC2:29</a>, <a class="ref" href="mesfunc1.html#T37" onmouseover="rs('mesfunc1/T37')" onmouseout="rh()">MESFUNC1:37</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E33:99"/>
 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="c11">C</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="c11">C</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="c14">I</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="c14">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</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">y</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="c17">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">J</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">y</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="c11">C</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">y</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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</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">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">y</font> ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">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="c14">I</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="c14">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( $1 <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">I</font> implies <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="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span></span>)</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="c18">x</font>)</span>)</span>) );<br/>
<a NAME="E1:99_10"/><span class="lab"><font color="Green" title="E83">I35</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T24" onmouseover="rs('finseq_3/T24')" onmouseout="rh()">FINSEQ_3:24</a></span>;<br/>
<a NAME="E2:99_10"/><span class="lab"><font color="Green" title="E84">I36</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_10_1"/><span class="lab"><font color="Green" title="E85">I37</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:99_10_1"/><span class="lab"><font color="Green" title="E86">I38</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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">I</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)</span><br/>

<a NAME="E3:99_10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">I39</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">I</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:99_10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">I40</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E5:99_10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">I41</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_10_1_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="E1:99_10_1_1"/>
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_10_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_10_1_1_1"/><span class="lab"><font color="Green" title="E90">I42</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)</span><br/><div class="add"><a NAME="E2:99_10_1_1_1"/><span class="lab"><font color="Green" title="E91">I43</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_10_1"><span class="lab"><font color="Green" title="E87">I39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E4:99_10_1_1_1"/><span class="lab"><font color="Green" title="E92">I44</font></span>: 
( <font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &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  <font color="Maroon" title="c20">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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> 1</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:99_10_1_1_1"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_10_1_1_1"><span class="lab"><font color="Green" title="E91">I43</font></span></a>, <a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:99_10_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">I46</font></span>: 
1 <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">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">I47</font></span>: <font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> 1 = 
<font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_10_1_1_1"><span class="lab"><font color="Green" title="E93">I46</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a></span>
;<br/><a NAME="E8:99_10_1_1_1"/>
<font color="Maroon" title="c8">F</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_10_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a>, <a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a></span>;<br/><a NAME="E9:99_10_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T20" onmouseover="rs('finseq_5/T20')" onmouseout="rh()">FINSEQ_5:20</a></span>;<br/><a NAME="E10:99_10_1_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" onmouseover="rs('finseq_1/T39')" onmouseout="rh()">FINSEQ_1:39</a></span>;<br/><a NAME="E11:99_10_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">I49</font></span>: 
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> 1 <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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span></span>)</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99_10_1_1_1"><span class="lab"><font color="Green" title="E94">I47</font></span></a>, <a class="ref" href="zfmisc_1.html#T25" onmouseover="rs('zfmisc_1/T25')" onmouseout="rh()">ZFMISC_1:25</a></span>;<br/><a NAME="E12:99_10_1_1_1"/>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_10_1_1_1"><span class="lab"><font color="Green" title="E90">I42</font></span></a>, <a class="txt" href="#E4:99_10_1_1_1"><span class="lab"><font color="Green" title="E92">I44</font></span></a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/><span class="kw">hence </span><a NAME="E13:99_10_1_1_1"/>
<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="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_10_1_1_1"><span class="lab"><font color="Green" title="E92">I44</font></span></a>, <a class="txt" href="#E11:99_10_1_1_1"><span class="lab"><font color="Green" title="E95">I49</font></span></a>, <a class="txt" href="#E1:99_10_1_1_1"><span class="lab"><font color="Green" title="E90">I42</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_10_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_10_1_1_2"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)</span><br/><div class="add"><a NAME="E2:99_10_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">I50</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E3:99_10_1_1_2"/>
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_10_1"><span class="lab"><font color="Green" title="E88">I40</font></span></a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/><a NAME="E4:99_10_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">I51</font></span>: 
<font color="Maroon" title="c19">k</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> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_10_1_1_2"><span class="lab"><font color="Green" title="E90">I50</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">I52</font></span>: <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="c14">I</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font> = 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</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="c18">x</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_10_1"><span class="lab"><font color="Green" title="E85">I37</font></span></a>, <a class="txt" href="#E2:99_10_1_1_2"><span class="lab"><font color="Green" title="E90">I50</font></span></a>, <a class="txt" href="#E4:99_10_1"><span class="lab"><font color="Green" title="E88">I40</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c19">k</font></span>)</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="c18">x</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>
;<br/><a NAME="E6:99_10_1_1_2"/><span class="lab"><font color="Green" title="E93">I53</font></span>: 
<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="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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">I</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_10_1"><span class="lab"><font color="Green" title="E87">I39</font></span></a>, <a class="txt" href="#E2:99_10_1_1_2"><span class="lab"><font color="Green" title="E90">I50</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/><a NAME="E7:99_10_1_1_2"/>
 <a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font> is  <a href="measur11.html#V2" title="MEASUR11:attr.2">without_-infty-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:99"><span class="lab"><font color="Green" title="E76">I15</font></span></a>, <a class="ref" href="measur11.html#T61" target="_self" onmouseover="rs('measur11/T61')" onmouseout="rh()">Th57</a></span>;<br/><a NAME="E8:99_10_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_10_1_1_2"><span class="lab"><font color="Green" title="E90">I50</font></span></a>, <a class="txt" href="#E5:99_10_1"><span class="lab"><font color="Green" title="E89">I41</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:99_10_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">I54</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">V178</a>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_10_1_1_2"><span class="lab"><font color="Green" title="E91">I51</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E10:99_10_1_1_2"/>
<font color="Maroon" title="c14">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:99"><span class="lab"><font color="Green" title="E75">I7</font></span></a>, <a class="txt" href="#E2:99_10_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a></span>;<br/><a NAME="E11:99_10_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">V178</a>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_10_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><span class="kw">then </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><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">I</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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">I</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_10_1_1_2"><span class="lab"><font color="Green" title="E94">I54</font></span></a>, <a class="ref" href="mesfunc5.html#T16" onmouseover="rs('mesfunc5/T16')" onmouseout="rh()">MESFUNC5:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">X1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">X1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">X1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c1">X1</font>

;<br/><a NAME="E13:99_10_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">I55</font></span>: 
<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="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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">I</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_10_1_1_2"><span class="lab"><font color="Green" title="E93">I53</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">E1</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c19">k</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T75" target="_self" onmouseover="rs('measur11/T75')" onmouseout="rh()">Th71</a></span>;<br/><a NAME="E15:99_10_1_1_2"/><span class="lab"><font color="Green" title="E96">I56</font></span>: 
( <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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">C</font> &amp; <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c8">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E2:99_10_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a>, <a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">E2</font> = <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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="#E4:99"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="finseq_2.html#T11" onmouseover="rs('finseq_2/T11')" onmouseout="rh()">FINSEQ_2:11</a></span>;<br/><span class="lab"><font color="Green" title="E97">I57</font></span>: <font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:99_10_1_1_2"><span class="lab"><font color="Green" title="E96">I56</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">E2</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E15:99_10_1_1_2"><span class="lab"><font color="Green" title="E96">I56</font></span></a></span>
;<br/><a NAME="E18:99_10_1_1_2"/>
<font color="Maroon" title="c14">I</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_10_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">I58</font></span>: <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="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> = 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p2">(<span class="default"><a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p3">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c20">E1</font>,<span class="p4"><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="c18">x</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p2">(<span class="default"><a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c18">x</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99"><span class="lab"><font color="Green" title="E74">I6</font></span></a>, <a class="txt" href="#E2:99_10_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a>, <a class="txt" href="#E5:99_10_1_1_2"><span class="lab"><font color="Green" title="E92">I52</font></span></a>, <a class="txt" href="#E13:99_10_1_1_2"><span class="lab"><font color="Green" title="E95">I55</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c20">E1</font>,<font color="Maroon" title="c18">x</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p2">(<span class="default"><a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p3">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">E2</font>,<span class="p4"><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="c18">x</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99_10_1_1_2"><span class="lab"><font color="Green" title="E97">I57</font></span></a>, <a class="ref" href="measur11.html#T72" target="_self" onmouseover="rs('measur11/T72')" onmouseout="rh()">Th68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c20">E1</font>,<font color="Maroon" title="c18">x</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c21">E2</font>,<font color="Maroon" title="c18">x</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T72" target="_self" onmouseover="rs('measur11/T72')" onmouseout="rh()">Th68</a></span>
;<br/><a NAME="E20:99_10_1_1_2"/>
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E21:99_10_1_1_2"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T1" target="_self" onmouseover="rs('measur11/T1')" onmouseout="rh()">Th72</a></span>;<br/><a NAME="E22:99_10_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">I59</font></span>: 
<font color="Maroon" title="c20">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c21">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><a NAME="E23:99_10_1_1_2"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T3" target="_self" onmouseover="rs('measur11/T3')" onmouseout="rh()">Th74</a></span>;<br/><a NAME="E24:99_10_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">I60</font></span>: 
<font color="Maroon" title="c20">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c21">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">E3</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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> ;<br/><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c20">E1</font>,<font color="Maroon" title="c18">x</font>)</span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p2">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c21">E2</font>,<font color="Maroon" title="c18">x</font>)</span>)</span></span>)</span> = 
<font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c22">E3</font>,<font color="Maroon" title="c18">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:99_10_1_1_2"><span class="lab"><font color="Green" title="E100">I60</font></span></a>, <a class="ref" href="measur11.html#T26" target="_self" onmouseover="rs('measur11/T26')" onmouseout="rh()">Th20</a></span>
<br/>.= 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T72" target="_self" onmouseover="rs('measur11/T72')" onmouseout="rh()">Th68</a></span>
;<br/><span class="kw">hence </span><a NAME="E27:99_10_1_1_2"/>
<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="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:99_10_1_1_2"><span class="lab"><font color="Green" title="E99">I59</font></span></a>, <a class="txt" href="#E19:99_10_1_1_2"><span class="lab"><font color="Green" title="E98">I58</font></span></a>, <a class="ref" href="measur11.html#T35" target="_self" onmouseover="rs('measur11/T35')" onmouseout="rh()">Th29</a>, <a class="ref" href="measure1.html#T30" onmouseover="rs('measure1/T30')" onmouseout="rh()">MEASURE1:30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:99_10"/><span class="lab"><font color="Green" title="E85">I61</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:99_10"><span class="lab"><font color="Green" title="E83">I35</font></span></a>, <a class="txt" href="#E2:99_10"><span class="lab"><font color="Green" title="E84">I36</font></span></a>);<br/></span>
<a NAME="E4:99_10"/><span class="lab"><font color="Green" title="E86">I62</font></span>: 
<font color="Maroon" title="c14">I</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E73">I3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:99_10"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">I</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">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" onmouseover="rs('finseq_5/T6')" onmouseout="rh()">FINSEQ_5:6</a></span>;<br/>
<a NAME="E6:99_10"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">I</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:99_10"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">I</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/>
<a NAME="E8:99_10"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">I</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> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c14">I</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E87">I63</font></span>: <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="c14">I</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="c14">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font> = 
<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="c14">I</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">I</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c6">M2</font>,<span class="p1">(<span class="default"><a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">I</font></span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_10"><span class="lab"><font color="Green" title="E85">I61</font></span></a>, <a class="txt" href="#E4:99_10"><span class="lab"><font color="Green" title="E86">I62</font></span></a>, <a class="ref" href="finseq_5.html#T6" onmouseover="rs('finseq_5/T6')" onmouseout="rh()">FINSEQ_5:6</a></span>
;<br/>
<font color="Maroon" title="c7">E</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</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">I</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:99"><span class="lab"><font color="Green" title="E82">K1</font></span></a>, <a class="ref" href="finseq_1.html#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:99_10"/>
 <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">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="c14">I</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="c14">I</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E9:99_10"><span class="lab"><font color="Green" title="E87">I63</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E34:99"/>
 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="c17">J</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">J</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="c11">C</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:99"><span class="lab"><font color="Green" title="E79">J6</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">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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</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">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">y</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E35:99"/>
 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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</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> 
  <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">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   <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">y</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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>  st <font color="Maroon" title="c18">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="c17">J</font> holds <br/><font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</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">let </span><font color="Maroon" title="c19">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="c18">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="c17">J</font> implies <font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:99_11"/><span class="lab"><font color="Green" title="E83">I16</font></span>: 
<font color="Maroon" title="c18">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="c17">J</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">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="E3:99_11"/><span class="lab"><font color="Green" title="E84">I17</font></span>: 
( <font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</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> holds  <font color="Maroon" title="c20">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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a></span>;<br/>
<a NAME="E4:99_11"/><span class="lab"><font color="Green" title="E85">I18</font></span>: 
<font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c20">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_11"><span class="lab"><font color="Green" title="E83">I16</font></span></a>, <a class="txt" href="#E3:99_11"><span class="lab"><font color="Green" title="E84">I17</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E5:99_11"/><span class="lab"><font color="Green" title="E86">I19</font></span>: 
<font color="Maroon" title="c18">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="c8">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="txt" href="#E1:99_11"><span class="lab"><font color="Green" title="E83">I16</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:99_11"/><span class="kw">then </span>
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/>
<a NAME="E7:99_11"/><span class="kw">then </span>
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur10.html#D5" onmouseover="rs('measur10/D5')" onmouseout="rh()">MEASUR10:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">An</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c22">Bn</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E9:99_11"/><span class="lab"><font color="Green" title="E87">I20</font></span>: 
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c21">An</font>,<font color="Maroon" title="c22">Bn</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">Fn</font> = <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E5:99_11"><span class="lab"><font color="Green" title="E86">I19</font></span></a>, <a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_11_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="E1:99_11_1"/>
( <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">An</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">An</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_11_1_1"/><span class="lab"><font color="Green" title="E88">I21</font></span>: 
<font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">An</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><div class="add"><a NAME="E2:99_11_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="c20">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="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c22">Bn</font>,<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="99_11_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">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="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c24">x</font></span><br/>

<a NAME="E1:99_11_1_1_1"/><span class="lab"><font color="Green" title="E89">I22</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Maroon" title="c24">x</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_11"><span class="lab"><font color="Green" title="E84">I17</font></span></a></span>;<br/>
<a NAME="E2:99_11_1_1_1"/><span class="lab"><font color="Green" title="E90">I23</font></span>: 
<font color="Maroon" title="c18">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="c11">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="txt" href="#E1:99_11"><span class="lab"><font color="Green" title="E83">I16</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:99_11_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E4:99_11_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> <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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E2:99_11_1_1_1"><span class="lab"><font color="Green" title="E90">I23</font></span></a></span>;<br/>
<a NAME="E5:99_11_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">I24</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c23">Fn</font>,<font color="Maroon" title="c24">x</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c24">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_11"><span class="lab"><font color="Green" title="E87">I20</font></span></a>, <a class="txt" href="#E1:99_11_1_1_1"><span class="lab"><font color="Green" title="E89">I22</font></span></a>, <a class="ref" href="measur11.html#T69" target="_self" onmouseover="rs('measur11/T69')" onmouseout="rh()">Th65</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_11_1_1_1_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="E1:99_11_1_1_1_1"/>
( <font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">Bn</font> or  not <font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">Bn</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_11_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_11_1_1_1_1_1"/><span class="lab"><font color="Green" title="E92">I25</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">Bn</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c24">x</font></span><br/><div class="add"><a NAME="E2:99_11_1_1_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"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c23">Fn</font>,<font color="Maroon" title="c24">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_11"><span class="lab"><font color="Green" title="E87">I20</font></span></a>, <a class="txt" href="#E1:99_11_1_1"><span class="lab"><font color="Green" title="E88">I21</font></span></a>, <a class="ref" href="measur11.html#T22" target="_self" onmouseover="rs('measur11/T22')" onmouseout="rh()">Th16</a>, <a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/><a NAME="E3:99_11_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:99_11_1_1_1"><span class="lab"><font color="Green" title="E91">I24</font></span></a>, <a class="ref" href="xxreal_3.html#T81" onmouseover="rs('xxreal_3/T81')" onmouseout="rh()">XXREAL_3:81</a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_11_1_1_1_1_1"/>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_11_1_1_1_1_1"><span class="lab"><font color="Green" title="E92">I25</font></span></a>, <a class="ref" href="measur10.html#D7" onmouseover="rs('measur10/D7')" onmouseout="rh()">MEASUR10:def 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_11_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_11_1_1_1_1_2"/><span class="lab"><font color="Green" title="E92">I26</font></span>: 
 not <font color="Maroon" title="c24">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">Bn</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c24">x</font></span><br/><div class="add"><a NAME="E2:99_11_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c24">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/><a NAME="E3:99_11_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">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="txt" href="#E5:99_11_1_1_1"><span class="lab"><font color="Green" title="E91">I24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_11_1_1_1_1_2"/>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_11_1_1_1_1_2"><span class="lab"><font color="Green" title="E92">I26</font></span></a>, <a class="ref" href="measur10.html#D7" onmouseover="rs('measur10/D7')" onmouseout="rh()">MEASUR10:def 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E3:99_11_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="measur10.html#K5" title="MEASUR10:func.5">Xchi</a> (<font color="Maroon" title="c22">Bn</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="E4:99_11_1_1"/>
<font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">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="#E4:99_11"><span class="lab"><font color="Green" title="E85">I18</font></span></a>, <a class="ref" href="measur10.html#T32" onmouseover="rs('measur10/T32')" onmouseout="rh()">MEASUR10:32</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_11_1_2"/><span class="lab"><font color="Green" title="E88">I27</font></span>: 
<font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">An</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">P</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><div class="add"><a NAME="E2:99_11_1_2"/>
<font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">An</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="supinf_2.html#T51" onmouseover="rs('supinf_2/T51')" onmouseout="rh()">SUPINF_2:51</a></span>;<br/><a NAME="E3:99_11_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">An</font> <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="txt" href="#E1:99_11_1_2"><span class="lab"><font color="Green" title="E88">I27</font></span></a>, <a class="ref" href="xxreal_0.html#T14" onmouseover="rs('xxreal_0/T14')" onmouseout="rh()">XXREAL_0:14</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">r</font> = <font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c21">An</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/><a NAME="E5:99_11_1_2"/><span class="lab"><font color="Green" title="E89">I28</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E6:99_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">I29</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X2</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D6" onmouseover="rs('mesfunc1/D6')" onmouseout="rh()">MESFUNC1:def 6</a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E7:99_11_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>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font> holds <br/><font color="Maroon" title="c20">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"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</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="99_11_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">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="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font> implies <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_11_1_2_1"/>
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font></span><br/>

<a NAME="E2:99_11_1_2_1"/><span class="lab"><font color="Green" title="E91">I30</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font></span>)</span>,<font color="Maroon" title="c25">x</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_11"><span class="lab"><font color="Green" title="E84">I17</font></span></a></span>;<br/>
<a NAME="E3:99_11_1_2_1"/><span class="lab"><font color="Green" title="E92">I31</font></span>: 
<font color="Maroon" title="c18">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="c11">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="txt" href="#E1:99_11"><span class="lab"><font color="Green" title="E83">I16</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:99_11_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E5:99_11_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> <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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_11_1_2_1"><span class="lab"><font color="Green" title="E92">I31</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a></span>;<br/>
<a NAME="E6:99_11_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">I32</font></span>: 
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c23">Fn</font>,<font color="Maroon" title="c25">x</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c25">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_11"><span class="lab"><font color="Green" title="E87">I20</font></span></a>, <a class="txt" href="#E2:99_11_1_2_1"><span class="lab"><font color="Green" title="E91">I30</font></span></a>, <a class="ref" href="measur11.html#T69" target="_self" onmouseover="rs('measur11/T69')" onmouseout="rh()">Th65</a></span>;<br/>
<a NAME="E7:99_11_1_2_1"/><span class="lab"><font color="Green" title="E94">I33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c25">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_11_1_2"><span class="lab"><font color="Green" title="E90">I29</font></span></a>, <a class="ref" href="mesfunc1.html#D6" onmouseover="rs('mesfunc1/D6')" onmouseout="rh()">MESFUNC1:def 6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_11_1_2_1_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="E1:99_11_1_2_1_1"/>
( <font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">Bn</font> or  not <font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">Bn</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_11_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_11_1_2_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">Bn</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:99_11_1_2_1_1_1"/>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99_11_1_2_1"><span class="lab"><font color="Green" title="E94">I33</font></span></a>, <a class="txt" href="#E6:99_11_1_2_1"><span class="lab"><font color="Green" title="E93">I32</font></span></a>, <a class="txt" href="#E9:99_11"><span class="lab"><font color="Green" title="E87">I20</font></span></a>, <a class="ref" href="measur11.html#T22" target="_self" onmouseover="rs('measur11/T22')" onmouseout="rh()">Th16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_11_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_11_1_2_1_1_2"/>
 not <font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c22">Bn</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font></span><br/><div class="add"><a NAME="E2:99_11_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">I34</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c25">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/><a NAME="E3:99_11_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<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="c25">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_11_1_2"><span class="lab"><font color="Green" title="E90">I29</font></span></a>, <a class="ref" href="mesfunc1.html#D6" onmouseover="rs('mesfunc1/D6')" onmouseout="rh()">MESFUNC1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:99_11_1_2_1_1_2"/>
<font color="Maroon" title="c20">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c25">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_11_1_2_1_1_2"><span class="lab"><font color="Green" title="E95">I34</font></span></a>, <a class="txt" href="#E6:99_11_1_2_1"><span class="lab"><font color="Green" title="E93">I32</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E8:99_11_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c24">r</font> <a href="mesfunc1.html#K6" title="MESFUNC1:func.6">(#)</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c22">Bn</font>,<font color="Maroon" title="c2">X2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_11_1_2"><span class="lab"><font color="Green" title="E90">I29</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E9:99_11_1_2"/>
<font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c18">n</font> is <font color="Maroon" title="c19">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="#E4:99_11"><span class="lab"><font color="Green" title="E85">I18</font></span></a>, <a class="txt" href="#E5:99_11_1_2"><span class="lab"><font color="Green" title="E89">I28</font></span></a>, <a class="ref" href="mesfunc2.html#T29" onmouseover="rs('mesfunc2/T29')" onmouseout="rh()">MESFUNC2:29</a>, <a class="ref" href="mesfunc1.html#T37" onmouseover="rs('mesfunc1/T37')" onmouseout="rh()">MESFUNC1:37</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E36:99"/>
 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="c11">C</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="c11">C</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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( $1 <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="c17">J</font> implies <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="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span></span>)</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="c18">x</font>)</span>)</span>) );<br/>
<a NAME="E1:99_12"/><span class="lab"><font color="Green" title="E83">I35</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T24" onmouseover="rs('finseq_3/T24')" onmouseout="rh()">FINSEQ_3:24</a></span>;<br/>
<a NAME="E2:99_12"/><span class="lab"><font color="Green" title="E84">I36</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="99_12_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:99_12_1"/><span class="lab"><font color="Green" title="E85">I37</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:99_12_1"/><span class="lab"><font color="Green" title="E86">I38</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c17">J</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)</span><br/>

<a NAME="E3:99_12_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">I39</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:99_12_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">I40</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E5:99_12_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">I41</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_12_1_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="E1:99_12_1_1"/>
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_12_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:99_12_1_1_1"/><span class="lab"><font color="Green" title="E90">I42</font></span>: 
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)</span><br/><div class="add"><a NAME="E2:99_12_1_1_1"/><span class="lab"><font color="Green" title="E91">I43</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_12_1"><span class="lab"><font color="Green" title="E87">I39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">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="E4:99_12_1_1_1"/><span class="lab"><font color="Green" title="E92">I44</font></span>: 
( <font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &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  <font color="Maroon" title="c20">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>  <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="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> 1</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:99_12_1_1_1"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_12_1_1_1"><span class="lab"><font color="Green" title="E91">I43</font></span></a>, <a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:99_12_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">I46</font></span>: 
1 <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">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">I47</font></span>: <font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> 1 = 
<font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_12_1_1_1"><span class="lab"><font color="Green" title="E93">I46</font></span></a>, <a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a></span>
;<br/><a NAME="E8:99_12_1_1_1"/>
<font color="Maroon" title="c8">F</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_12_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a>, <a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a></span>;<br/><a NAME="E9:99_12_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T20" onmouseover="rs('finseq_5/T20')" onmouseout="rh()">FINSEQ_5:20</a></span>;<br/><a NAME="E10:99_12_1_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" onmouseover="rs('finseq_1/T39')" onmouseout="rh()">FINSEQ_1:39</a></span>;<br/><a NAME="E11:99_12_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">I49</font></span>: 
<font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> 1 <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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span></span>)</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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99_12_1_1_1"><span class="lab"><font color="Green" title="E94">I47</font></span></a>, <a class="ref" href="zfmisc_1.html#T25" onmouseover="rs('zfmisc_1/T25')" onmouseout="rh()">ZFMISC_1:25</a></span>;<br/><a NAME="E12:99_12_1_1_1"/>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_12_1_1_1"><span class="lab"><font color="Green" title="E90">I42</font></span></a>, <a class="txt" href="#E4:99_12_1_1_1"><span class="lab"><font color="Green" title="E92">I44</font></span></a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/><span class="kw">hence </span><a NAME="E13:99_12_1_1_1"/>
<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="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_12_1_1_1"><span class="lab"><font color="Green" title="E92">I44</font></span></a>, <a class="txt" href="#E11:99_12_1_1_1"><span class="lab"><font color="Green" title="E95">I49</font></span></a>, <a class="txt" href="#E1:99_12_1_1_1"><span class="lab"><font color="Green" title="E90">I42</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_12_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:99_12_1_1_2"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)</span><br/><div class="add"><a NAME="E2:99_12_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">I50</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E3:99_12_1_1_2"/>
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_12_1"><span class="lab"><font color="Green" title="E88">I40</font></span></a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/><a NAME="E4:99_12_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">I51</font></span>: 
<font color="Maroon" title="c19">k</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> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_12_1_1_2"><span class="lab"><font color="Green" title="E90">I50</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">I52</font></span>: <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="c17">J</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</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="c18">x</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99_12_1"><span class="lab"><font color="Green" title="E85">I37</font></span></a>, <a class="txt" href="#E2:99_12_1_1_2"><span class="lab"><font color="Green" title="E90">I50</font></span></a>, <a class="txt" href="#E4:99_12_1"><span class="lab"><font color="Green" title="E88">I40</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="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"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c19">k</font></span>)</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="c18">x</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>
;<br/><a NAME="E6:99_12_1_1_2"/><span class="lab"><font color="Green" title="E93">I53</font></span>: 
<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="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c17">J</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_12_1"><span class="lab"><font color="Green" title="E87">I39</font></span></a>, <a class="txt" href="#E2:99_12_1_1_2"><span class="lab"><font color="Green" title="E90">I50</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/><a NAME="E7:99_12_1_1_2"/>
 <a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font> is  <a href="measur11.html#V2" title="MEASUR11:attr.2">without_-infty-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:99"><span class="lab"><font color="Green" title="E81">J15</font></span></a>, <a class="ref" href="measur11.html#T61" target="_self" onmouseover="rs('measur11/T61')" onmouseout="rh()">Th57</a></span>;<br/><a NAME="E8:99_12_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_12_1_1_2"><span class="lab"><font color="Green" title="E90">I50</font></span></a>, <a class="txt" href="#E5:99_12_1"><span class="lab"><font color="Green" title="E89">I41</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:99_12_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">I54</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">V178</a>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:99_12_1_1_2"><span class="lab"><font color="Green" title="E91">I51</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E10:99_12_1_1_2"/>
<font color="Maroon" title="c17">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:99"><span class="lab"><font color="Green" title="E80">J7</font></span></a>, <a class="txt" href="#E2:99_12_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a></span>;<br/><a NAME="E11:99_12_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">V178</a>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_12_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><span class="kw">then </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><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="c17">J</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="mesfunc1.html#K3" title="MESFUNC1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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="c17">J</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:99_12_1_1_2"><span class="lab"><font color="Green" title="E94">I54</font></span></a>, <a class="ref" href="mesfunc5.html#T16" onmouseover="rs('mesfunc5/T16')" onmouseout="rh()">MESFUNC5:16</a></span>
<br/>.= 
<font color="Maroon" title="c2">X2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c2">X2</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">X2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c2">X2</font>

;<br/><a NAME="E13:99_12_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">I55</font></span>: 
<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="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c17">J</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_12_1_1_2"><span class="lab"><font color="Green" title="E93">I53</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">E1</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c19">k</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T75" target="_self" onmouseover="rs('measur11/T75')" onmouseout="rh()">Th71</a></span>;<br/><a NAME="E15:99_12_1_1_2"/><span class="lab"><font color="Green" title="E96">I56</font></span>: 
( <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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">C</font> &amp; <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c8">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E2:99_12_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a>, <a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">E2</font> = <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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="#E4:99"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="finseq_2.html#T11" onmouseover="rs('finseq_2/T11')" onmouseout="rh()">FINSEQ_2:11</a></span>;<br/><span class="lab"><font color="Green" title="E97">I57</font></span>: <font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c11">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:99_12_1_1_2"><span class="lab"><font color="Green" title="E96">I56</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">E2</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E15:99_12_1_1_2"><span class="lab"><font color="Green" title="E96">I56</font></span></a></span>
;<br/><a NAME="E18:99_12_1_1_2"/>
<font color="Maroon" title="c17">J</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_12_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">I58</font></span>: <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="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> = 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p3">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c20">E1</font>,<span class="p4"><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="c18">x</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c11">C</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c18">x</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:99"><span class="lab"><font color="Green" title="E79">J6</font></span></a>, <a class="txt" href="#E2:99_12_1"><span class="lab"><font color="Green" title="E86">I38</font></span></a>, <a class="txt" href="#E5:99_12_1_1_2"><span class="lab"><font color="Green" title="E92">I52</font></span></a>, <a class="txt" href="#E13:99_12_1_1_2"><span class="lab"><font color="Green" title="E95">I55</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c20">E1</font>,<font color="Maroon" title="c18">x</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p3">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c21">E2</font>,<span class="p4"><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="c18">x</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:99_12_1_1_2"><span class="lab"><font color="Green" title="E97">I57</font></span></a>, <a class="ref" href="measur11.html#T72" target="_self" onmouseover="rs('measur11/T72')" onmouseout="rh()">Th68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c20">E1</font>,<font color="Maroon" title="c18">x</font>)</span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c21">E2</font>,<font color="Maroon" title="c18">x</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T72" target="_self" onmouseover="rs('measur11/T72')" onmouseout="rh()">Th68</a></span>
;<br/><a NAME="E20:99_12_1_1_2"/>
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E21:99_12_1_1_2"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T1" target="_self" onmouseover="rs('measur11/T1')" onmouseout="rh()">Th72</a></span>;<br/><a NAME="E22:99_12_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">I59</font></span>: 
<font color="Maroon" title="c20">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c21">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><a NAME="E23:99_12_1_1_2"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T3" target="_self" onmouseover="rs('measur11/T3')" onmouseout="rh()">Th74</a></span>;<br/><a NAME="E24:99_12_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">I60</font></span>: 
<font color="Maroon" title="c20">E1</font> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <font color="Maroon" title="c21">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">E3</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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> ;<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="c20">E1</font>,<font color="Maroon" title="c18">x</font>)</span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p2">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c21">E2</font>,<font color="Maroon" title="c18">x</font>)</span>)</span></span>)</span> = 
<font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K10" title="MEASUR11:func.10">Measurable-Y-section</a> (<font color="Maroon" title="c22">E3</font>,<font color="Maroon" title="c18">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:99_12_1_1_2"><span class="lab"><font color="Green" title="E100">I60</font></span></a>, <a class="ref" href="measur11.html#T26" target="_self" onmouseover="rs('measur11/T26')" onmouseout="rh()">Th20</a></span>
<br/>.= 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T72" target="_self" onmouseover="rs('measur11/T72')" onmouseout="rh()">Th68</a></span>
;<br/><span class="kw">hence </span><a NAME="E27:99_12_1_1_2"/>
<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="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:99_12_1_1_2"><span class="lab"><font color="Green" title="E99">I59</font></span></a>, <a class="txt" href="#E19:99_12_1_1_2"><span class="lab"><font color="Green" title="E98">I58</font></span></a>, <a class="ref" href="measur11.html#T35" target="_self" onmouseover="rs('measur11/T35')" onmouseout="rh()">Th29</a>, <a class="ref" href="measure1.html#T30" onmouseover="rs('measure1/T30')" onmouseout="rh()">MEASURE1:30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:99_12"/><span class="lab"><font color="Green" title="E85">I61</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:99_12"><span class="lab"><font color="Green" title="E83">I35</font></span></a>, <a class="txt" href="#E2:99_12"><span class="lab"><font color="Green" title="E84">I36</font></span></a>);<br/></span>
<a NAME="E4:99_12"/><span class="lab"><font color="Green" title="E86">I62</font></span>: 
<font color="Maroon" title="c17">J</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E17:99"><span class="lab"><font color="Green" title="E78">J3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:99_12"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</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="c17">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" onmouseover="rs('finseq_5/T6')" onmouseout="rh()">FINSEQ_5:6</a></span>;<br/>
<a NAME="E6:99_12"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:99_12"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/>
<a NAME="E8:99_12"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</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> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c17">J</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E87">I63</font></span>: <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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font> = 
<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="c17">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="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"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p4">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</font></span>)</span></span>)</span></span>)</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="c18">x</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_12"><span class="lab"><font color="Green" title="E85">I61</font></span></a>, <a class="txt" href="#E4:99_12"><span class="lab"><font color="Green" title="E86">I62</font></span></a>, <a class="ref" href="finseq_5.html#T6" onmouseover="rs('finseq_5/T6')" onmouseout="rh()">FINSEQ_5:6</a></span>
;<br/>
<font color="Maroon" title="c7">E</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">J</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:99"><span class="lab"><font color="Green" title="E82">K1</font></span></a>, <a class="ref" href="finseq_1.html#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:99_12"/>
 <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="c11">C</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="c11">C</font></span>)</span></span>)</span>,<font color="Maroon" title="c18">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="c17">J</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="c17">J</font></span>)</span></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c18">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E9:99_12"><span class="lab"><font color="Green" title="E87">I63</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
