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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">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> holds <br/> ( (  for <font color="Olive" title="b4">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Olive" title="b3">E</font>,<font color="Olive" title="b4">p</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">S2</font> ) &amp; (  for <font color="Olive" title="b4">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Olive" title="b3">E</font>,<font color="Olive" title="b4">p</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">S1</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">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> holds <br/> ( (  for <font color="Olive" title="b3">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Olive" title="b2">E</font>,<font color="Olive" title="b3">p</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">S2</font> ) &amp; (  for <font color="Olive" title="b3">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Olive" title="b2">E</font>,<font color="Olive" title="b3">p</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</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">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> holds <br/> ( (  for <font color="Olive" title="b2">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">p</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font> ) &amp; (  for <font color="Olive" title="b2">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">p</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c5">E</font>,<font color="Olive" title="b1">p</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font> ) &amp; (  for <font color="Olive" title="b1">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c5">E</font>,<font color="Olive" title="b1">p</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</font> ) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c6">K</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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> :  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font> </span> } </span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">K</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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> :  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font> </span> } </span>  as   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <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="ref" href="measur11.html#T47" target="_self" onmouseover="rs('measur11/T47')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E2:57"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
(  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> &amp;  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T47" target="_self" onmouseover="rs('measur11/T47')" onmouseout="rh()">Th42</a>, <a class="ref" href="srings_3.html#T21" onmouseover="rs('srings_3/T21')" onmouseout="rh()">SRINGS_3:21</a></span>;<br/>
<a NAME="E3:57"/><span class="kw">then </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> <font color="Maroon" title="c7">K</font>
 
;<br/>
<a NAME="E4:57"/><span class="kw">then </span>
 <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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">K</font>
 
<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="E5:57"/><span class="kw">then </span>
<font color="Maroon" title="c5">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">K</font>
 
;<br/>
<a NAME="E6:57"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">C</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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> st <br/>( <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">C</font> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E7:57"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<font color="Maroon" title="c5">E</font>,<font color="Olive" title="b1">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <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">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c5">E</font>,<font color="Olive" title="b1">p</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">K2</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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> :  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</font> </span> } </span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">K2</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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> :  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</font> </span> } </span>  as   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <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="ref" href="measur11.html#T48" target="_self" onmouseover="rs('measur11/T48')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E9:57"/>
 <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T48" target="_self" onmouseover="rs('measur11/T48')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E10:57"/><span class="kw">then </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> <font color="Maroon" title="c9">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E44">A1</font></span></a></span>;<br/>
<a NAME="E11:57"/><span class="kw">then </span>
 <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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">K2</font>
 
<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="E12:57"/><span class="kw">then </span>
<font color="Maroon" title="c5">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">K2</font>
 
;<br/>
<a NAME="E13:57"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">C</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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> st <br/>( <font color="Maroon" title="c5">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">C</font> &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S1</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E14:57"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="measur11.html#K8" title="MEASUR11:func.8">Y-section</a> (<font color="Maroon" title="c5">E</font>,<font color="Olive" title="b1">x</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <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"> verum</span><br/>


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