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<div class="add">

<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">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">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="c4">S</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c3">Si</font>;<br/>







<b>assume </b><a NAME="E1:97"/><i><font color="Green" title="E60">A1</font></i>: 
( <font color="Maroon" title="c4">S</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp;  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_value_of</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ;<br/>

<a NAME="E2:97"/>
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="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp;  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_value_of</a> <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> holds <br/>(  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <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> <font color="Maroon" title="c2">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" title="c5">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/>

<b>assume </b><a NAME="E1:97_1"/><i><font color="Green" title="E61">A2</font></i>: 
( <font color="Maroon" title="c5">A1</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp;  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_value_of</a> <font color="Maroon" title="c5">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 ;<br/>

<a NAME="E2:97_1"/><i><font color="Green" title="E62">A3</font></i>: 
 <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_inf</a> <font color="Maroon" title="c5">A1</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"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c5">A1</font></span>)</span>
 
;<br/>
<a NAME="E3:97_1"/>
 <a href="kurato_2.html#K5" title="KURATO_2:func.5">lim_sup</a> <font color="Maroon" title="c5">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="c5">A1</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:97_1"/>
(  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c5">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <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="c5">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="setlim_1.html#E1:97_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="setlim_1.html#E2:97_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="setlim_1.html#T64" target="_self" title="SETLIM_1:th.64">Th64</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:97"/><b>then </b><i><font color="Green" title="E61">A4</font></i>: 
(  <a href="setlim_1.html#K6" title="SETLIM_1:func.6">lim_inf</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="setlim_1.html#K7" title="SETLIM_1:func.7">lim_sup</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="setlim_1.html#E1:97"><i><font color="Green" title="E60">A1</font></i></a></i>;<br/>
<a NAME="E4:97"/><b>then </b>
<font color="Maroon" title="c4">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="E5:97"/>
( <font color="Maroon" title="c4">S</font> is <a href="setlim_1.html#V3" title="SETLIM_1:attr.3">convergent</a> &amp;  <a href="setlim_1.html#K8" title="SETLIM_1:func.8">lim</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="setlim_1.html#K6" title="SETLIM_1:func.6">lim_inf</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="setlim_1.html#K7" title="SETLIM_1:func.7">lim_sup</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="setlim_1.html#E3:97"><i><font color="Green" title="E61">A4</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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