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<span class="kw">let </span><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S1</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</font><br/>  for <font color="Olive" title="b2">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b3">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S1</font><br/>  for <font color="Olive" title="b4">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b2">S2</font><br/>  for <font color="Olive" title="b5">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="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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b5">F</font> is    <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="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Olive" title="b3">M1</font>,<font color="Olive" title="b4">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Olive" title="b5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b3">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Olive" title="b5">F</font></span>)</span>,<font color="Olive" title="b4">M2</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S1</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b2">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b3">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">S2</font><br/>  for <font color="Olive" title="b4">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="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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b4">F</font> is    <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="Olive" title="b1">S2</font>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Olive" title="b2">M1</font>,<font color="Olive" title="b3">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Olive" title="b4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b2">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Olive" title="b4">F</font></span>)</span>,<font color="Olive" title="b3">M2</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">S2</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M1</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font><br/>  for <font color="Olive" title="b2">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font><br/>  for <font color="Olive" title="b3">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="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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b3">F</font> is    <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>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Olive" title="b1">M1</font>,<font color="Olive" title="b2">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Olive" title="b1">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Olive" title="b3">F</font></span>)</span>,<font color="Olive" title="b2">M2</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">M1</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c3">S1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M2</font> being   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font><br/>  for <font color="Olive" title="b2">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="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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b2">F</font> is    <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>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Olive" title="b1">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Olive" title="b2">F</font></span>)</span>,<font color="Olive" title="b1">M2</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">M2</font> be   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c4">S2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Olive" title="b1">F</font> is    <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>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Olive" title="b1">F</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c7">F</font> be   <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="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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Maroon" title="c7">F</font> is    <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>) holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a>  &amp; <font color="Maroon" title="c7">F</font> is    <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>) implies <span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>) )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:131"/><span class="lab"><font color="Green" title="E96">A1</font></span>: 
<font color="Maroon" title="c6">M2</font> is  <a href="measur11.html#V4" title="MEASUR11:attr.4">sigma_finite</a> 
 <span class="kw">and </span><br/><a NAME="E2:131"/><span class="lab"><font color="Green" title="E97">A2</font></span>: 
<font color="Maroon" title="c7">F</font> is    <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>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span><br/>

<a NAME="E3:131"/><span class="lab"><font color="Green" title="E98">A3</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>);<br/>
<a NAME="E4:131"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T2" onmouseover="rs('zfmisc_1/T2')" onmouseout="rh()">ZFMISC_1:2</a></span>;<br/>
<a NAME="E5:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A4</font></span>: 
 <a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<a NAME="E6:131"/>
 not  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T24" onmouseover="rs('finseq_3/T24')" onmouseout="rh()">FINSEQ_3:24</a></span>;<br/>
<a NAME="E7:131"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>;<br/>
<a NAME="E8:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">P1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:131"><span class="lab"><font color="Green" title="E99">A4</font></span></a>, <a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E2:131"><span class="lab"><font color="Green" title="E97">A2</font></span></a>, <a class="ref" href="measur11.html#T100" target="_self" onmouseover="rs('measur11/T100')" onmouseout="rh()">Th99</a></span>;<br/>
<a NAME="E9:131"/><span class="lab"><font color="Green" title="E101">P2</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="131_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:131_1"/><span class="lab"><font color="Green" title="E102">P3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">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">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:131_1"/><span class="lab"><font color="Green" title="E103">A6</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="131_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:131_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="131_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:131_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><a NAME="E2:131_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T55" onmouseover="rs('finseq_3/T55')" onmouseout="rh()">FINSEQ_3:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:131_1"><span class="lab"><font color="Green" title="E103">A6</font></span></a>, <a class="ref" href="finseq_1.html#T82" onmouseover="rs('finseq_1/T82')" onmouseout="rh()">FINSEQ_1:82</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T112" onmouseover="rs('finseq_3/T112')" onmouseout="rh()">FINSEQ_3:112</a></span>
;<br/><span class="kw">then </span> <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T31" onmouseover="rs('finseq_1/T31')" onmouseout="rh()">FINSEQ_1:31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a></span>
;<br/><span class="kw">then </span> <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T78" onmouseover="rs('zfmisc_1/T78')" onmouseout="rh()">ZFMISC_1:78</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T25" onmouseover="rs('zfmisc_1/T25')" onmouseout="rh()">ZFMISC_1:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>
;<br/><a NAME="E6:131_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A8</font></span>: 
 <a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T4" target="_self" onmouseover="rs('measur11/T4')" onmouseout="rh()">Th101</a>, <a class="ref" href="measur11.html#T18" target="_self" onmouseover="rs('measur11/T18')" onmouseout="rh()">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p3">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p2">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E2:131"><span class="lab"><font color="Green" title="E97">A2</font></span></a>, <a class="txt" href="#E1:131_1"><span class="lab"><font color="Green" title="E102">P3</font></span></a>, <a class="ref" href="measur11.html#T100" target="_self" onmouseover="rs('measur11/T100')" onmouseout="rh()">Th99</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:131_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E96">A1</font></span></a>, <a class="txt" href="#E6:131_1_1_1"><span class="lab"><font color="Green" title="E104">A8</font></span></a>, <a class="ref" href="measur11.html#T4" target="_self" onmouseover="rs('measur11/T4')" onmouseout="rh()">Th101</a>, <a class="ref" href="measur11.html#T95" target="_self" onmouseover="rs('measur11/T95')" onmouseout="rh()">Th94</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="131_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:131_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><a NAME="E2:131_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T49" onmouseover="rs('finseq_3/T49')" onmouseout="rh()">FINSEQ_3:49</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:131_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_1"><span class="lab"><font color="Green" title="E102">P3</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E10:131"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E8:131"><span class="lab"><font color="Green" title="E100">P1</font></span></a>, <a class="txt" href="#E9:131"><span class="lab"><font color="Green" title="E101">P2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E11:131"/>
<span class="p1">(<span class="default"><a href="measur11.html#K6" title="MEASUR11:func.6">product_sigma_Measure</a> (<font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c6">M2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<font color="Maroon" title="c5">M1</font>,<span class="p1">(<span class="default"><a href="measur11.html#K19" title="MEASUR11:func.19">Y-vol</a> (<span class="p2">(<span class="default"><a href="measur11.html#K2" title="MEASUR11:func.2">Union</a> <font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c6">M2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:131"><span class="lab"><font color="Green" title="E98">A3</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>
