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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">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S1</font><br/>  for <font color="Olive" title="b6">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">S2</font> holds  <span class="p1">(<span class="default"><a href="measur11.html#K5" title="MEASUR11:func.5">product_Measure</a> (<font color="Olive" title="b3">M1</font>,<font color="Olive" title="b4">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b5">A</font>,<font color="Olive" title="b6">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">A</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">S1</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b2">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b3">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">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b5">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S2</font> holds  <span class="p1">(<span class="default"><a href="measur11.html#K5" title="MEASUR11:func.5">product_Measure</a> (<font color="Olive" title="b2">M1</font>,<font color="Olive" title="b3">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b4">A</font>,<font color="Olive" title="b5">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">A</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">S2</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b2">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">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b4">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <span class="p1">(<span class="default"><a href="measur11.html#K5" title="MEASUR11:func.5">product_Measure</a> (<font color="Olive" title="b1">M1</font>,<font color="Olive" title="b2">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">A</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">M1</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b3">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <span class="p1">(<span class="default"><a href="measur11.html#K5" title="MEASUR11:func.5">product_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Olive" title="b1">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <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> <font color="Olive" title="b2">A</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">M2</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <span class="p1">(<span class="default"><a href="measur11.html#K5" title="MEASUR11:func.5">product_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1"><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> <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> <font color="Olive" title="b1">A</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> holds  <span class="p1">(<span class="default"><a href="measur11.html#K5" title="MEASUR11:func.5">product_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Olive" title="b1">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">B</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">B</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measur11.html#K5" title="MEASUR11:func.5">product_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">B</font></span>)</span></span><br/>













<span class="kw">set </span><font color="Maroon" title="c9">S</font> =  <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>);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">P</font> =  <a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">m</font> =  <a href="measur11.html#K5" title="MEASUR11:func.5">product_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>);<br/>
<a NAME="E1:17"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 <a href="srings_3.html#K2" title="SRINGS_3:func.2">DisUnion</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="srings_3.html#T22" onmouseover="rs('srings_3/T22')" onmouseout="rh()">SRINGS_3:22</a></span>;<br/>
<a NAME="E2:17"/>
<span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> : verum </span> } </span> 
 
;<br/>
<a NAME="E3:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur10.html#D5" onmouseover="rs('measur10/D5')" onmouseout="rh()">MEASUR10:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">F</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as   <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>) <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T74" onmouseover="rs('finseq_1/T74')" onmouseout="rh()">FINSEQ_1:74</a></span>;<br/>
<a NAME="E5:17"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
 <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="srings_3.html#K2" title="SRINGS_3:func.2">DisUnion</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="srings_3.html#T12" onmouseover="rs('srings_3/T12')" onmouseout="rh()">SRINGS_3:12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">SumPF</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E7:17"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
(  <a href="extreal1.html#K2" title="EXTREAL1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">SumPF</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</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"><span class="p3">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">SumPF</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &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="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"><span class="p2">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> holds <br/><font color="Maroon" title="c13">SumPF</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c13">SumPF</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">n</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"><span class="p3">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#D2" onmouseover="rs('extreal1/D2')" onmouseout="rh()">EXTREAL1:def 2</a></span>;<br/>
<a NAME="E8:17"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<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="E9:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A6</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="c12">F</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="E10:17"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="finseq_3.html#T120" onmouseover="rs('finseq_3/T120')" onmouseout="rh()">FINSEQ_3:120</a></span>;<br/>
<a NAME="E11:17"/><span class="kw">then </span>
 <a href="extreal1.html#K2" title="EXTREAL1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">SumPF</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></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"><span class="p3">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17"><span class="lab"><font color="Green" title="E15">A4</font></span></a></span>;<br/>
<a NAME="E12:17"/><span class="kw">then </span>
 <a href="extreal1.html#K2" title="EXTREAL1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</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="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="xxreal_3.html#T4" onmouseover="rs('xxreal_3/T4')" onmouseout="rh()">XXREAL_3:4</a></span>;<br/>
<a NAME="E13:17"/><span class="kw">then </span>
 <a href="extreal1.html#K2" title="EXTREAL1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>;<br/>
<a NAME="E14:17"/><span class="kw">then </span>
 <a href="extreal1.html#K2" title="EXTREAL1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
;<br/>
<a NAME="E15:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A7</font></span>: 
 <a href="extreal1.html#K2" title="EXTREAL1:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur10.html#K6" title="MEASUR10:func.6">product-pre-Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur10.html#T22" onmouseover="rs('measur10/T22')" onmouseout="rh()">MEASUR10:22</a></span>;<br/>
<a NAME="E16:17"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></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="E17:17"/><span class="kw">then </span>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T25" onmouseover="rs('zfmisc_1/T25')" onmouseout="rh()">ZFMISC_1:25</a></span>;<br/>
<a NAME="E18:17"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="prob_3.html#K5" title="PROB_3:func.5">Union</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></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">hence </span><a NAME="E19:17"/>
<span class="p1">(<span class="default"><a href="measur11.html#K5" title="MEASUR11:func.5">product_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">M2</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E15:17"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="measure9.html#D8" onmouseover="rs('measure9/D8')" onmouseout="rh()">MEASURE9:def 8</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>
