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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">F1</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Olive" title="b1">S1</font><br/>  for <font color="Olive" title="b4">F2</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Olive" title="b2">S2</font><br/>  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <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="Olive" title="b3">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>)</span>)</span></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">F1</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b3">F2</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Olive" title="b1">S2</font><br/>  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <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="Olive" title="b2">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Olive" title="b1">S2</font>)</span>)</span></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">F1</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b2">F2</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c4">S2</font><br/>  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <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="Olive" title="b1">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">F1</font> be    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</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">F2</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c4">S2</font><br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <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="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">F2</font> be    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <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="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">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"> <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="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span></span><br/>











<span class="kw">set </span><font color="Maroon" title="c8">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/>
<a NAME="E1:15"/>
( <font color="Maroon" title="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</font> &amp; <font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" onmouseover="rs('measure8/D2')" onmouseout="rh()">MEASURE8:def 2</a></span>;<br/>
<a NAME="E2:15"/><span class="kw">then </span>
<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="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></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="E3:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<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="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></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/>
<a NAME="E4:15"/><span class="lab"><font color="Green" title="E11">A2</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/>
<a NAME="E5:15"/>
 <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="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="srings_3.html#K2" title="SRINGS_3:func.2">DisUnion</a> <span class="p2">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D9" onmouseover="rs('prob_1/D9')" onmouseout="rh()">PROB_1:def 9</a></span>;<br/>
<a NAME="E6:15"/><span class="kw">then </span>
<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="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="srings_3.html#K2" title="SRINGS_3:func.2">DisUnion</a> <span class="p2">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E11">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:15"/>
<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="c5">F1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T7" target="_self" onmouseover="rs('measur11/T7')" onmouseout="rh()">Th1</a></span>; <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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