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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">F</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Olive" title="b1">F</font> iff for <font color="Olive" title="b3">n</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>   ex <font color="Olive" title="b4">k</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>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b4">k</font></span>)</span> )</span><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 <font color="Maroon" title="c1">X</font>; <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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c2">F</font> iff for <font color="Olive" title="b2">n</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>   ex <font color="Olive" title="b3">k</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>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">k</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c2">F</font> iff for <font color="Olive" title="b1">n</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>   ex <font color="Olive" title="b2">k</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>  st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">k</font></span>)</span> )</span><br/>





<div><b>hereby </b> <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">n</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>   ex <font color="Olive" title="b2">k</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>  st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">k</font></span>)</span> ) implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c2">F</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E7">A1</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#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c2">F</font>
 ; <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">n</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>   ex <font color="Olive" title="b2">k</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>  st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">k</font></span>)</span></span><br/><b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:10_1"/><i><font color="Green" title="E8">A2</font></i>: 
(  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c4">f</font> &amp; ( for <font color="Olive" title="b1">n</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">f</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="kurato_2.html#D4" target="_self" title="KURATO_2:def.4">Def4</a></i>;<br/><b>let </b><font color="Maroon" title="c5">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k</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>  st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span></span><br/><b>set </b><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c2">F</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font>;<br/><a NAME="E4:10_1"/>
<font color="Maroon" title="c4">f</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:10_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/><a NAME="E5:10_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:10_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="#E3:10_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="kurato_2.html#T6" target="_self" title="KURATO_2:th.6">Th6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">k</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> <b> such that </b><br/><a NAME="E7:10_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/><b>take </b><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c7">k</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> <font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span></span><br/><b>thus </b><a NAME="E8:10_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:10_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</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>assume </b><a NAME="E2:10"/><i><font color="Green" title="E7">A4</font></i>: 
for <font color="Olive" title="b1">n</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>   ex <font color="Olive" title="b2">k</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>  st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">k</font></span>)</span>
 ; <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#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c2">F</font></span><br/>

<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:10"/><i><font color="Green" title="E8">A5</font></i>: 
(  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c4">f</font> &amp; ( for <font color="Olive" title="b1">n</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">f</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Olive" title="b1">n</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="kurato_2.html#D4" target="_self" title="KURATO_2:def.4">Def4</a></i>;<br/>
<a NAME="E5:10"/>
for <font color="Olive" title="b1">z</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">z</font> be    <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> ; <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> <font color="Maroon" title="c4">f</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">z</font></span><br/>

<b>set </b><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c2">F</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">z</font>;<br/>
<a NAME="E1:10_2"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:10"><i><font color="Green" title="E8">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">k</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> <b> such that </b><br/><a NAME="E3:10_2"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:10"><i><font color="Green" title="E7">A4</font></i></a></i>;<br/>
<a NAME="E4:10_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="valued_0.html#K1" title="VALUED_0:func.1">^\</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:10_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="txt" href="#E1:10_2"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="#E3:10_2"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</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="E6:10"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K4" title="KURATO_2:func.4">lim_sup</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E4:10"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="kurato_2.html#T6" target="_self" title="KURATO_2:th.6">Th6</a></i>; <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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