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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">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="b4">E</font>, <font color="Olive" title="b5">V</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><br/>  for <font color="Olive" title="b6">P</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</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><br/>  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>  st <font color="Olive" title="b6">P</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b6">P</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">E</font> holds <br/> ex <font color="Olive" title="b8">K</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b2">S2</font> st <br/>( <font color="Olive" title="b8">K</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b9">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b8">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Olive" title="b6">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b9">n</font></span>)</span>,<font color="Olive" title="b7">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b5">V</font>,<font color="Olive" title="b7">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b8">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b4">E</font>,<font color="Olive" title="b7">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b5">V</font>,<font color="Olive" title="b7">x</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">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="b3">E</font>, <font color="Olive" title="b4">V</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><br/>  for <font color="Olive" title="b5">P</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</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><br/>  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>  st <font color="Olive" title="b5">P</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b5">P</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">E</font> holds <br/> ex <font color="Olive" title="b7">K</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">S2</font> st <br/>( <font color="Olive" title="b7">K</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b7">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Olive" title="b5">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b8">n</font></span>)</span>,<font color="Olive" title="b6">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b4">V</font>,<font color="Olive" title="b6">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b7">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b3">E</font>,<font color="Olive" title="b6">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b4">V</font>,<font color="Olive" title="b6">x</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">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>, <font color="Olive" title="b3">V</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><br/>  for <font color="Olive" title="b4">P</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</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/>  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>  st <font color="Olive" title="b4">P</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b4">P</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">E</font> holds <br/> ex <font color="Olive" title="b6">K</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c4">S2</font> st <br/>( <font color="Olive" title="b6">K</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b6">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Olive" title="b4">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b7">n</font></span>)</span>,<font color="Olive" title="b5">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b3">V</font>,<font color="Olive" title="b5">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b2">E</font>,<font color="Olive" title="b5">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b3">V</font>,<font color="Olive" title="b5">x</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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>, <font color="Olive" title="b2">V</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><br/>  for <font color="Olive" title="b3">P</font> being    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</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/>  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>  st <font color="Olive" title="b3">P</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b3">P</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b1">E</font> holds <br/> ex <font color="Olive" title="b5">K</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c4">S2</font> st <br/>( <font color="Olive" title="b5">K</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b5">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Olive" title="b3">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b6">n</font></span>)</span>,<font color="Olive" title="b4">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b2">V</font>,<font color="Olive" title="b4">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b1">E</font>,<font color="Olive" title="b4">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Olive" title="b2">V</font>,<font color="Olive" title="b4">x</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">E</font>, <font color="Maroon" title="c7">V</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="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</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/>  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>  st <font color="Olive" title="b1">P</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">P</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">E</font> holds <br/> ex <font color="Olive" title="b3">K</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c4">S2</font> st <br/>( <font color="Olive" title="b3">K</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b3">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Olive" title="b2">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c6">E</font>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Olive" title="b2">x</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">P</font> be    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</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">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="Maroon" title="c8">P</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c8">P</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">E</font> holds <br/> ex <font color="Olive" title="b2">K</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c4">S2</font> st <br/>( <font color="Olive" title="b2">K</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b2">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Olive" title="b1">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c6">E</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Olive" title="b1">x</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">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="c8">P</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c8">P</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">E</font> implies  ex <font color="Olive" title="b1">K</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c4">S2</font> st <br/>( <font color="Olive" title="b1">K</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> ) )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:139"/><span class="lab"><font color="Green" title="E104">A1</font></span>: 
<font color="Maroon" title="c8">P</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a> 
 <span class="kw">and </span><br/><a NAME="E2:139"/><span class="lab"><font color="Green" title="E105">A2</font></span>: 
 <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c8">P</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">E</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">K</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c4">S2</font> st <br/>( <font color="Olive" title="b1">K</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> )</span><br/>

<a NAME="E3:139"/><span class="lab"><font color="Green" title="E106">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c8">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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>
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">P1</font> = <font color="Maroon" title="c8">P</font> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</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> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">G</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c2">X2</font><span class="kw"> such that </span><br/><a NAME="E6:139"/><span class="lab"><font color="Green" title="E107">A5</font></span>: 
( <font color="Maroon" title="c11">G</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c11">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">P1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c9">x</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:139"><span class="lab"><font color="Green" title="E104">A1</font></span></a>, <a class="ref" href="measur11.html#T45" target="_self" onmouseover="rs('measur11/T45')" onmouseout="rh()">Th39</a></span>;<br/>
<a NAME="E7:139"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c11">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="139_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="c11">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font></span><br/>

<a NAME="E1:139_1"/>
<font color="Maroon" title="c10">P1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="txt" href="#E3:139"><span class="lab"><font color="Green" title="E106">A4</font></span></a></span>;<br/>
<a NAME="E2:139_1"/><span class="kw">then </span>
 <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">P1</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c9">x</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="measur11.html#T49" target="_self" onmouseover="rs('measur11/T49')" onmouseout="rh()">Th44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:139_1"/>
<font color="Maroon" title="c11">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">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="txt" href="#E6:139"><span class="lab"><font color="Green" title="E107">A5</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">G</font> = <font color="Maroon" title="c11">G</font> as    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <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/>
<span class="kw">set </span><font color="Maroon" title="c13">K</font> = <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c12">G</font>;<br/>
<a NAME="E9:139"/><span class="lab"><font color="Green" title="E108">A6</font></span>: 
( <font color="Maroon" title="c12">G</font> is  <a href="kurato_0.html#V3" title="KURATO_0:attr.3">convergent</a>  &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K3" title="PROB_1:func.3">Intersection</a> <font color="Maroon" title="c12">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:139"><span class="lab"><font color="Green" title="E107">A5</font></span></a>, <a class="ref" href="setlim_1.html#T64" onmouseover="rs('setlim_1/T64')" onmouseout="rh()">SETLIM_1:64</a></span>;<br/>
<a NAME="E10:139"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">P</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:139"><span class="lab"><font color="Green" title="E107">A5</font></span></a>, <a class="ref" href="measur11.html#T31" target="_self" onmouseover="rs('measur11/T31')" onmouseout="rh()">Th25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E109">A7</font></span>:  <a href="prob_1.html#K3" title="PROB_1:func.3">Intersection</a> <font color="Maroon" title="c12">G</font> = 
 <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p2">(<span class="default"><a href="measure3.html#K1" title="MEASURE3:func.1">rng</a> <font color="Maroon" title="c8">P</font></span>)</span></span>)</span>,<font color="Maroon" title="c9">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#T8" onmouseover="rs('setlim_1/T8')" onmouseout="rh()">SETLIM_1:8</a></span>
<br/>.= 
 <a href="measur11.html#K7" title="MEASUR11:func.7">X-section</a> (<span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Intersection</a> <font color="Maroon" title="c8">P</font></span>)</span>,<font color="Maroon" title="c9">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_1.html#T8" onmouseover="rs('setlim_1/T8')" onmouseout="rh()">SETLIM_1:8</a></span>
<br/>.= 
 <a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:139"><span class="lab"><font color="Green" title="E104">A1</font></span></a>, <a class="txt" href="#E2:139"><span class="lab"><font color="Green" title="E105">A2</font></span></a>, <a class="ref" href="setlim_1.html#T64" onmouseover="rs('setlim_1/T64')" onmouseout="rh()">SETLIM_1:64</a></span>
;<br/>
<a NAME="E12:139"/><span class="lab"><font color="Green" title="E110">A8</font></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"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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/>
<a NAME="E13:139"/>
 for <font color="Olive" title="b1">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/><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="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="139_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  implies <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="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c12">G</font></span>)</span> <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> <font color="Maroon" title="c4">S2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:139_3"/>
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">(<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="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c12">G</font></span>)</span> <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> <font color="Maroon" title="c4">S2</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">n1</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>  ;<br/>
<a NAME="E3:139_3"/>
<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="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c15">n1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1: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="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" onmouseover="rs('setlim_2/D5')" onmouseout="rh()">SETLIM_2:def 5</a></span>;<br/>
<a NAME="E4:139_3"/><span class="kw">then </span>
<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="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c15">n1</font></span>)</span>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:139"><span class="lab"><font color="Green" title="E107">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:139_3"/>
<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="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c12">G</font></span>)</span> <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> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">K</font> = <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="setlim_2.html#K5" title="SETLIM_2:func.5">(/\)</a> <font color="Maroon" title="c12">G</font> as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c4">S2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:139"><span class="lab"><font color="Green" title="E110">A8</font></span></a>, <a class="ref" href="funct_2.html#T3" onmouseover="rs('funct_2/T3')" onmouseout="rh()">FUNCT_2:3</a></span>;<br/>
<a NAME="E15:139"/><span class="lab"><font color="Green" title="E111">A9</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c14">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="139_4"><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="c14">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c15">n</font></span>)</span>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span></span><br/>

<a NAME="E1:139_4"/>
<font color="Maroon" title="c14">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1: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="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setlim_2.html#D5" onmouseover="rs('setlim_2/D5')" onmouseout="rh()">SETLIM_2:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:139_4"/>
<font color="Maroon" title="c14">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Maroon" title="c15">n</font></span>)</span>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:139"><span class="lab"><font color="Green" title="E107">A5</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">take </span>
<font color="Maroon" title="c14">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" title="c14">K</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c14">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c14">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E16:139"/>
( <font color="Maroon" title="c14">K</font> is  <a href="prob_1.html#V2" title="PROB_1:attr.2">non-ascending</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c14">K</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="measur11.html#K17" title="MEASUR11:func.17">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> ) &amp;  <a href="setlim_1.html#NK7" title="SETLIM_1:NK.7">lim</a> <font color="Maroon" title="c14">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> <a href="measure1.html#K2" title="MEASURE1:func.2">/\</a> <span class="p1">(<span class="default"><a href="measur11.html#K9" title="MEASUR11:func.9">Measurable-X-section</a> (<font color="Maroon" title="c7">V</font>,<font color="Maroon" title="c9">x</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:139"><span class="lab"><font color="Green" title="E111">A9</font></span></a>, <a class="txt" href="#E11:139"><span class="lab"><font color="Green" title="E109">A7</font></span></a>, <a class="txt" href="#E9:139"><span class="lab"><font color="Green" title="E108">A6</font></span></a>, <a class="txt" href="#E6:139"><span class="lab"><font color="Green" title="E107">A5</font></span></a>, <a class="ref" href="setlim_2.html#T21" onmouseover="rs('setlim_2/T21')" onmouseout="rh()">SETLIM_2:21</a>, <a class="ref" href="setlim_2.html#T92" onmouseover="rs('setlim_2/T92')" onmouseout="rh()">SETLIM_2:92</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>
