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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 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="c1">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> ) implies  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c1">A</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">B</font> )</span><br/>

<b>assume </b><a NAME="E1:89"/><i><font color="Green" title="E51">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="c1">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c1">A</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">B</font></span><br/>

<a NAME="E2:89"/>
 <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c1">A</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">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">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="c1">A</font> or <font color="Maroon" title="c3">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">B</font> )</span><br/>

<b>assume </b><a NAME="E1:89_1"/>
<font color="Maroon" title="c3">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="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">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">B</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">A1</font> being    <a href="kurato_2.html#M1" title="KURATO_2:mode.1">subsequence</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:89_1"/><i><font color="Green" title="E52">A2</font></i>: 
<font color="Maroon" title="c3">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="c4">A1</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>consider </b><font color="Maroon" title="c5">D</font> being    <a href="kurato_2.html#M1" title="KURATO_2:mode.1">subsequence</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E5:89_1"/><i><font color="Green" title="E53">A3</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="c4">A1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">D</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:89"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="kurato_2.html#T69" target="_self" title="KURATO_2:th.69">Th69</a></i>;<br/>
<a NAME="E6:89_1"/>
 <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c4">A1</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="c5">D</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E5:89_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="kurato_2.html#T54" target="_self" title="KURATO_2:th.54">Th54</a></i>;<br/>
<b>hence </b><a NAME="E7:89_1"/>
<font color="Maroon" title="c3">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">B</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E3:89_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="kurato_2.html#D12" target="_self" title="KURATO_2:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:89"/>
 <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c1">A</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">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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