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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>





<b>assume </b><a NAME="E1:95"/><i><font color="Green" title="E56">A1</font></i>: 
for <font color="Olive" title="b1">i</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="c2">F</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>thus </b><a NAME="E2:95"/>
 <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#T68" target="_self" title="KURATO_2:th.68">Th68</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E3:95"/>
 <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c2">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:95_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">G</font> being    <a href="kurato_2.html#M1" title="KURATO_2:mode.1">subsequence</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E3:95_1"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#D12" target="_self" title="KURATO_2:def.12">Def12</a></i>;<br/>
<b>thus </b><a NAME="E4:95_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:95"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="kurato_2.html#E3:95_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="kurato_2.html#T47" target="_self" title="KURATO_2:th.47">Th47</a></i>;<br/>


</div><b>end;</b></div>


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