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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/>





<b>assume </b><a NAME="E1:100"/>
<font color="Maroon" title="c3">S</font> is <a href="prob_1.html#V3" title="PROB_1:attr.3">non-decreasing</a>
 ;<br/>

<a NAME="E2:100"/><b>then </b><i><font color="Green" title="E67">A1</font></i>: 
(  <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> &amp;  <a href="setlim_1.html#K6" title="SETLIM_1:func.6">lim_inf</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="ref" href="setlim_1.html#T86" target="_self" title="SETLIM_1:th.86">Th86</a>, <a class="ref" href="setlim_1.html#T87" target="_self" title="SETLIM_1:th.87">Th87</a></i>;<br/>
<b>hence </b><a NAME="E3:100"/>
<font color="Maroon" title="c3">S</font> is <a href="setlim_1.html#V3" title="SETLIM_1:attr.3">convergent</a>
 <b>by </b><i><a class="ref" href="setlim_1.html#D12" target="_self" title="SETLIM_1:def.12">Def12</a></i>;<br/>

<b>hence </b><a NAME="E4:100"/>
 <a href="setlim_1.html#K8" title="SETLIM_1:func.8">lim</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#E2:100"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="setlim_1.html#D13" target="_self" title="SETLIM_1:def.13">Def13</a></i>;<br/>


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