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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">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A2</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/><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="E1:71"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">A2</font></span>)</span>
 ;<br/>

<a NAME="E2:71"/><b>then </b><i><font color="Green">E25</font></i>: 
( <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> &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">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>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="E4:71"/><i><font color="Green">E26</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">A2</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="setlim_2.html#D7" target="_self">Def7</a>, <a class="ref" href="kurato_2.html#T8">KURATO_2:8</a></i>;<br/>
<b>assume </b><a NAME="E5:71"/>
not <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">A1</font> <a href="setlim_2.html#K3" target="_self">(\)</a> <font color="Maroon">A2</font></span>)</span>
 ;<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="E7:71"/><i><font color="Green">E33</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> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="setlim_2.html#K3" target="_self">(\)</a> <font color="Maroon">A2</font></span>)</span> <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#T8">KURATO_2:8</a></i>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:71_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="setlim_2.html#K3" target="_self">(\)</a> <font color="Maroon">A2</font></span>)</span> <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">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="setlim_2.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E2:71_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</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">n2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:71_1"/>
(  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">k</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</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">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<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="E10:71"/><i><font color="Green">E36</font></i>: 
<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"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <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#D7" target="_self">Def7</a>, <a class="ref" href="kurato_2.html#T8">KURATO_2:8</a></i>;<br/>
<a NAME="E11:71"/>
( <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"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k1</font></span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</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> <span class="p2">(<span class="default"><font color="Maroon">n2</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#D8" target="_self">Def8</a>, </i>;<br/>
<a NAME="E12:71"/><b>then </b>
( <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> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span></span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</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> <span class="p2">(<span class="default"><font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E13:71"/>
contradiction
 <b>by </b><i/>;<br/>


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