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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Si</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">S</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Si</font>;<br/>





<a NAME="E1:98"/><i><font color="Green" title="E65">A1</font></i>: 
for <font color="Olive" title="b1">A1</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A1</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> holds <br/> <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">superior_setsequence</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Olive" title="b1">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E1:98_1"/>
 <a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">superior_setsequence</a> <font color="Maroon" title="c4">A1</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E2:98_1"/>
( <font color="Maroon" title="c4">A1</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> implies  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">superior_setsequence</a> <font color="Maroon" title="c4">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c4">A1</font> )
 <b>by </b><i><a class="ref" href="setlim_1.html#T65" target="_self" title="SETLIM_1:th.65">Th65</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:98"/>
( <font color="Maroon" title="c3">S</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a> implies  <a href="setlim_1.html#K7" title="SETLIM_1:func.7">lim_sup</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <font color="Maroon" title="c3">S</font> )
 <b>by </b><i><a class="txt" href="setlim_1.html#E1:98"><i><font color="Green" title="E65">A1</font></i></a></i>;<br/>


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