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<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">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">S</font>;<br/>







<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E5">AS0</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/>

<a NAME="E2:4"/><i><font color="Green" title="E6">B1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K5" title="SETLIM_1:func.5">@superior_setsequence</a> <font color="Maroon" title="c4">F</font></span>)</span> <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="E3:4"/>
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/><span class="p1">(<span class="default"><a href="setlim_1.html#K5" title="SETLIM_1:func.5">@superior_setsequence</a> <font color="Maroon" title="c4">F</font></span>)</span> <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/>
<b>then reconsider </b><font color="Maroon" title="c5">G</font> =  <a href="setlim_1.html#K5" title="SETLIM_1:func.5">@superior_setsequence</a> <font color="Maroon" title="c4">F</font> as   <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="txt" href="mesfunc8.html#E2:4"><i><font color="Green" title="E6">B1</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<a NAME="E5:4"/><i><font color="Green" title="E7">P1</font></i>: 
for <font color="Olive" title="b1">n</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="c5">G</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">n</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="E1:4_1"/><i><font color="Green" title="E8">P11</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E2:4_1"/>
 <a href="setlim_1.html#K5" title="SETLIM_1:func.5">@superior_setsequence</a> <font color="Maroon" title="c4">F</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a>
 
<b>by </b><i><a class="ref" href="setlim_1.html#T72" title="SETLIM_1:th.72">SETLIM_1:72</a></i>;<br/>
<b>hence </b><a NAME="E3:4_1"/>
<font color="Maroon" title="c5">G</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D6" title="PROB_1:def.6">PROB_1:def 6</a>, <a class="txt" href="mesfunc8.html#E1:4_1"><i><font color="Green" title="E8">P11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:4"/>
<font color="Maroon" title="c5">G</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="setlim_1.html#D3" title="SETLIM_1:def.3">SETLIM_1:def 3</a></i>;<br/>
<a NAME="E7:4"/><b>then </b>
<font color="Maroon" title="c5">G</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="setlim_1.html#T9" title="SETLIM_1:th.9">SETLIM_1:9</a></i>;<br/>
<a NAME="E8:4"/><b>then </b><i><font color="Green" title="E8">P2</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc8.html#E1:4"><i><font color="Green" title="E5">AS0</font></i></a>, <a class="txt" href="mesfunc8.html#E5:4"><i><font color="Green" title="E7">P1</font></i></a>, <a class="ref" href="measure3.html#T14" title="MEASURE3:th.14">MEASURE3:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">F1</font> = <font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c7">G1</font> = <font color="Maroon" title="c5">G</font> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E11:4"/><i><font color="Green" title="E9">P3</font></i>: 
(  <a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <font color="Maroon" title="c6">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c8">f</font> &amp; ( for <font color="Olive" title="b1">n</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="c8">f</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F1</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Olive" title="b1">n</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="kurato_2.html#D4" title="KURATO_2:def.4">KURATO_2:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">n</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="E1:4_2"/>
<font color="Maroon" title="c8">f</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F1</font> <a href="kurato_2.html#K3" title="KURATO_2:func.3">^\</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc8.html#E11:4"><i><font color="Green" title="E9">P3</font></i></a></i>;<br/><b>hence </b><a NAME="E2:4_2"/>
<font color="Maroon" title="c8">f</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="ref" href="mesfunc8.html#T2" target="_self" title="MESFUNC8:th.2">Lm410</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:4"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">G1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E14:4"/>
 ex <font color="Olive" title="b1">G</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> ,<font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="setlim_1.html#K5" title="SETLIM_1:func.5">@superior_setsequence</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c3">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K7" title="SETLIM_1:func.7">lim_sup</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">G</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc8.html#E8:4"><i><font color="Green" title="E8">P2</font></i></a>, <a class="txt" href="mesfunc8.html#E11:4"><i><font color="Green" title="E9">P3</font></i></a></i>;<br/>


</div>
