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<span class="kw">consider </span><font color="Maroon" title="c7">IT</font> being   <a href="measure1.html#NM1" title="MEASURE1:NM.1">Measure</a> of <span class="p1">(<span class="default"><a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p2">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:12_1_1"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 for <font color="Olive" title="b1">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> holds <br/> for <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>)  st <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> holds <br/><font color="Maroon" title="c7">IT</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure9.html#T55" onmouseover="rs('measure9/T55')" onmouseout="rh()">MEASURE9:55</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">IT</font> = <font color="Maroon" title="c7">IT</font> as    <a href="measure9.html#M2" title="MEASURE9:mode.2">induced_Measure</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>), <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="measure9.html#D8" onmouseover="rs('measure9/D8')" onmouseout="rh()">MEASURE9:def 8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">IT</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="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> holds <br/> for <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>)  st <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> holds <br/><font color="Maroon" title="c8">IT</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b2">F</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E4:12_1_1"/>
 for <font color="Olive" title="b1">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> holds <br/> for <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>)  st <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> holds <br/><font color="Maroon" title="c8">IT</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E8">A1</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/>


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