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





<b>assume </b><a NAME="E1:89"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A1</font> is <a href="kurato_2.html#V3">convergent</a>
 ;<br/>

<b>thus </b><a NAME="E2:89"/>
 <a href="kurato_2.html#K5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K9" target="_self">(\+\)</a> <font color="Maroon">A1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K5">\+\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">A1</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K9" target="_self">(\+\)</a> <font color="Maroon">A1</font></span>)</span>
 ;<br/>

<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:89_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K9" target="_self">(\+\)</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span>
 <b>by </b><i><a class="ref" href="setlim_2.html#T31" target="_self">Th31</a>, <a class="ref" href="kurato_2.html#T8">KURATO_2:8</a></i>;<br/><a NAME="E3:89_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K5">\+\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="setlim_2.html#T34" target="_self">Th34</a>, </i>;<br/><a NAME="E4:89_1_1"/><b>then </b>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#T1">XBOOLE_0:1</a></i>;<br/><b>hence </b><a NAME="E5:89_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span> ) )
 ;<br/></div><b>end;</b></div>
<b>assume </b><a NAME="E3:89_1"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K5">\+\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">A1</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2"/>
( ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">A1</font> ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">A1</font> ) )
 </span><b>by </b><i><a class="ref" href="setlim_2.html#T34" target="_self">Th34</a>, <a class="ref" href="xboole_0.html#T1">XBOOLE_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1"><b>suppose </b></a><a NAME="E1:89_1_2_1"/><i><font color="Green">E37</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:89_1_2_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="kurato_2.html#T8">KURATO_2:8</a></i>;<br/><b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:89_1_2_1"/><i><font color="Green">E40</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="setlim_2.html#T33" target="_self">Th33</a></i>;<br/><b>thus </b><a NAME="E6:89_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="setlim_2.html#T36" target="_self">Th36</a>, <a class="ref" href="setlim_2.html#T37" target="_self">Th37</a>, <a class="ref" href="setlim_2.html#T35" target="_self">Th35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_2"><b>suppose </b></a><a NAME="E1:89_1_2_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><a NAME="E2:89_1_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="setlim_2.html#T30" target="_self">Th30</a>, <a class="ref" href="kurato_2.html#D7">KURATO_2:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:89_1_2_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="kurato_2.html#T7">KURATO_2:7</a></i>;<br/><b>consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:89_1_2_2"/><i><font color="Green">E44</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="setlim_2.html#T33" target="_self">Th33</a></i>;<br/><b>thus </b><a NAME="E7:89_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="setlim_2.html#T36" target="_self">Th36</a>, <a class="ref" href="setlim_2.html#T37" target="_self">Th37</a>, <a class="ref" href="setlim_2.html#T35" target="_self">Th35</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E3:89"/>
<font color="Maroon">A</font> <a href="subset_1.html#K5">\+\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">A1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K5">lim_sup</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="setlim_2.html#K9" target="_self">(\+\)</a> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="setlim_2.html#T25" target="_self">Th25</a></i>;<br/>


</div>
