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<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Si</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Si</font>;<br/>







<b>assume </b><a NAME="E1:97"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">S</font> is <a href="funct_1.html#V3">constant</a> &amp;  <a href="yellow_6.html#K1">the_value_of</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E2:97"/>
for <font color="Olive">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>  st <font color="Olive">A1</font> is <a href="funct_1.html#V3">constant</a> &amp;  <a href="yellow_6.html#K1">the_value_of</a> <font color="Olive">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> holds <br/>(  <a href="kurato_2.html#K1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Olive">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Olive">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:97_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A1</font> is <a href="funct_1.html#V3">constant</a> &amp;  <a href="yellow_6.html#K1">the_value_of</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E2:97_1"/><i><font color="Green">E30</font></i>: 
 <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span>
 
;<br/>
<a NAME="E3:97_1"/>
 <a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:97_1"/>
(  <a href="kurato_2.html#K1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="setlim_1.html#T19" target="_self">Th19</a>, <a class="ref" href="setlim_1.html#T20" target="_self">Th20</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:97"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="setlim_1.html#K6" target="_self">lim_inf</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="setlim_1.html#K7" target="_self">lim_sup</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="setlim_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E4:97"/><b>then </b>
<font color="Maroon">S</font> is <a href="setlim_1.html#V3" target="_self">convergent</a>
 
<b>by </b><i><a class="ref" href="setlim_1.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E5:97"/>
( <font color="Maroon">S</font> is <a href="setlim_1.html#V3" target="_self">convergent</a> &amp;  <a href="setlim_1.html#K8" target="_self">lim</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="setlim_1.html#K6" target="_self">lim_inf</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="setlim_1.html#K7" target="_self">lim_sup</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="setlim_1.html#T19" target="_self">Th19</a>, <a class="ref" href="setlim_1.html#T13" target="_self">Th13</a></i>;<br/>


</div>
