<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be  <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="mesfunc8.html#M1" title="MESFUNC8:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>;<br/>







<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E30">AS</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> &amp; ( for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c4">E</font> ) )
 ;<br/>

<a NAME="E2:36"/><i><font color="Green" title="E31">L0</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim_sup</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc8.html#D9" target="_self" title="MESFUNC8:def.9">deflimsupf</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> = <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K4" title="MESFUNC8:func.4">superior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> $1</span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c6">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:36_1"/><i><font color="Green" title="E32">AS0</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c6">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E3:36_1"/><i><font color="Green" title="E33">AS1</font></i>: 
for <font color="Olive" title="b1">x</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c6">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K4" title="MESFUNC8:func.4">superior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E2:36_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K4" title="MESFUNC8:func.4">superior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc8.html#E2:36_1"><i><font color="Green" title="E32">AS0</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:36_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K4" title="MESFUNC8:func.4">superior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:36_1_2"/><i><font color="Green" title="E34">Q1</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K4" title="MESFUNC8:func.4">superior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E2:36_1"><i><font color="Green" title="E32">AS0</font></i></a></i>;<br/><a NAME="E2:36_1_2"/><i><font color="Green" title="E35">Q2</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc8.html#K4" title="MESFUNC8:func.4">superior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 <b>by </b><i><a class="ref" href="mesfunc8.html#T19" target="_self" title="MESFUNC8:th.19">Lm4202x</a>, <a class="txt" href="mesfunc8.html#E1:36"><i><font color="Green" title="E30">AS</font></i></a></i>;<br/><a NAME="E3:36_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc8.html#K4" title="MESFUNC8:func.4">superior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="mesfunc5.html#K4" title="MESFUNC5:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc8.html#D6" target="_self" title="MESFUNC8:def.6">defsup</a></i>;<br/><a NAME="E4:36_1_2"/><b>then </b>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfunc8.html#K4" title="MESFUNC8:func.4">superior_realsequence</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E1:36"><i><font color="Green" title="E30">AS</font></i></a>, <a class="ref" href="mesfunc1.html#T31" title="MESFUNC1:th.31">MESFUNC1:31</a>, <a class="txt" href="mesfunc8.html#E2:36_1_2"><i><font color="Green" title="E35">Q2</font></i></a></i>;<br/><b>hence </b><a NAME="E5:36_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E1:36_1_2"><i><font color="Green" title="E34">Q1</font></i></a>, <a class="ref" href="mesfunc1.html#D11" title="MESFUNC1:def.11">MESFUNC1:def 11</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c6">F</font> as    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/><a NAME="E6:36_1"/><i><font color="Green" title="E34">L1</font></i>: 
 <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim_sup</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc8.html#T21" target="_self" title="MESFUNC8:th.21">Lm4202c</a>, <a class="txt" href="mesfunc8.html#E1:36"><i><font color="Green" title="E30">AS</font></i></a>, <a class="txt" href="mesfunc8.html#E3:36_1"><i><font color="Green" title="E33">AS1</font></i></a></i>;<br/><a NAME="E7:36_1"/><i><font color="Green" title="E35">P1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:36_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D9" title="PROB_1:def.9">PROB_1:def 9</a></i>;<br/><a NAME="E9:36_1"/><b>then </b><i><font color="Green" title="E36">T0</font></i>: 
<font color="Maroon" title="c7">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a>, <a class="txt" href="mesfunc8.html#E7:36_1"><i><font color="Green" title="E35">P1</font></i></a></i>;<br/><a NAME="E10:36_1"/><b>then </b><i><font color="Green" title="E37">T1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font> is   <a href="measure1.html#NM3" title="MEASURE1:NM.3">N_Sub_set_fam</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="measure1.html#T52" title="MEASURE1:th.52">MEASURE1:52</a></i>;<br/><a NAME="E11:36_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E9:36_1"><i><font color="Green" title="E36">T0</font></i></a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E12:36_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font> is    <a href="measure2.html#M1" title="MEASURE2:mode.1">N_Measure_fam</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E10:36_1"><i><font color="Green" title="E37">T1</font></i></a>, <a class="ref" href="measure2.html#D1" title="MEASURE2:def.1">MEASURE2:def 1</a></i>;<br/><a NAME="E13:36_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="measure2.html#T3" title="MEASURE2:th.3">MEASURE2:3</a></i>;<br/><b>hence </b><a NAME="E14:36_1"/>
<font color="Maroon" title="c4">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" title="MESFUNC1:func.17">great_eq_dom</a> <span class="p2">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim_sup</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E6:36_1"><i><font color="Green" title="E34">L1</font></i></a>, <a class="ref" href="mesfunc1.html#D11" title="MESFUNC1:def.11">MESFUNC1:def 11</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:36"/>
 <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim_sup</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc1.html#R2" title="MESFUNC1:pred.2">is_measurable_on</a> <font color="Maroon" title="c4">E</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#T31" title="MESFUNC1:th.31">MESFUNC1:31</a>, <a class="txt" href="mesfunc8.html#E2:36"><i><font color="Green" title="E31">L0</font></i></a>, <a class="txt" href="mesfunc8.html#E1:36"><i><font color="Green" title="E30">AS</font></i></a></i>;<br/>


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