<?xml version="1.0"?>
<div class="add">

<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>assume </b><a NAME="E1:68"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">A1</font> is <a href="kurato_2.html#V3">convergent</a> &amp; <font color="Maroon">A2</font> is <a href="kurato_2.html#V3">convergent</a> )
 ;<br/>

<b>thus </b><a NAME="E2:68"/>
 <a href="kurato_2.html#K4">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="setlim_2.html#K4" target="_self">(\+\)</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K5">\+\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="setlim_2.html#T14" target="_self">Th14</a>, <a class="ref" href="setlim_2.html#T12" target="_self">Th12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E3:68"/>
<span class="p1">(<span class="default"><a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K5">\+\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K4">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="setlim_2.html#K4" target="_self">(\+\)</a> <font color="Maroon">A2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_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="E1:68_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K5">\+\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A2</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1"/>
( ( <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> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A2</font> ) or ( not <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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A2</font> ) )
 </span><b>by </b><i><a class="ref" href="setlim_2.html#D5" target="_self">Def5</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="68_1_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_1"/><i><font color="Green">E33</font></i>: 
( <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> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A2</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="E3:68_1_1_1"/><i><font color="Green">E34</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">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#T7">KURATO_2:7</a></i>;<br/><a NAME="E4:68_1_1_1"/>
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="setlim_2.html#D6" target="_self">Def6</a>, <a class="ref" href="setlim_2.html#T14" target="_self">Th14</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="E6:68_1_1_1"/><i><font color="Green">E36</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">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><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1_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="E1:68_1_1_1_1"/>
( <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> <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> &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">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="setlim_2.html#D7" target="_self">Def7</a>, <a class="ref" href="setlim_2.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E2:68_1_1_1_1"/><b>then </b>
<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"><span class="p3">(<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">k</font></span>)</span></span>)</span> <a href="subset_1.html#K5">\+\</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"><span class="p3">(<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">k</font></span>)</span></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="E3:68_1_1_1_1"/>
<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#K4" target="_self">(\+\)</a> <font color="Maroon">A2</font></span>)</span> <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">k</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:68_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="setlim_2.html#K4" target="_self">(\+\)</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="kurato_2.html#T7">KURATO_2:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_2"/><i><font color="Green">E37</font></i>: 
( not <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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A2</font> )
 ;<br/><div class="add"><a NAME="E2:68_1_1_2"/><b>then </b>
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>
 <b>by </b><i><a class="ref" href="setlim_2.html#T14" target="_self">Th14</a>, <a class="ref" href="kurato_2.html#D7">KURATO_2:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">n3</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="E4:68_1_1_2"/><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">n3</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">n4</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="E6:68_1_1_2"/><i><font color="Green">E40</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">A2</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n4</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#D6" target="_self">Def6</a>, <a class="ref" href="kurato_2.html#T7">KURATO_2:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_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="E1:68_1_1_2_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">n3</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n4</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> &amp; <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">n4</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="setlim_2.html#D7" target="_self">Def7</a>, <a class="ref" href="setlim_2.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E2:68_1_1_2_1"/><b>then </b>
<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"><span class="p3">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n4</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="subset_1.html#K5">\+\</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"><span class="p3">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n4</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></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="E3:68_1_1_2_1"/>
<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#K4" target="_self">(\+\)</a> <font color="Maroon">A2</font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n3</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n4</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:68_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="setlim_2.html#K4" target="_self">(\+\)</a> <font color="Maroon">A2</font></span>)</span>
 <b>by </b><i><a class="ref" href="kurato_2.html#T7">KURATO_2:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div>
