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

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</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="c3">C</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) implies  <a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c2">B</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:74"/><i><font color="Green" title="E39">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="c3">C</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/>

<a NAME="E2:74"/><i><font color="Green" title="E40">A2</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="c3">C</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</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">C</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><br/>

<a NAME="E1:74_1"/>
<font color="Maroon" title="c3">C</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:74"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:74_1"/>
<font color="Maroon" title="c3">C</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</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>
<a NAME="E3:74"/><i><font color="Green" title="E41">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="c3">C</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</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="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</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">C</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><br/>

<a NAME="E1:74_2"/>
<font color="Maroon" title="c3">C</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:74"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:74_2"/>
<font color="Maroon" title="c3">C</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</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>let </b><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c3">C</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c2">B</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E4:74"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c5">C1</font> being    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c3">C</font><b> such that </b><br/><a NAME="E6:74"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K6" title="KURATO_2:func.6">Lim_inf</a> <font color="Maroon" title="c5">C1</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="c6">D1</font> being    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E8:74"/><i><font color="Green" title="E43">A5</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="c5">C1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">D1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <b>by </b><i><a class="txt" href="#E2:74"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="kurato_2.html#T67" target="_self" title="KURATO_2:th.67">Th69</a></i>;<br/>
<a NAME="E9:74"/>
 <a href="kurato_2.html#K6" title="KURATO_2:func.6">Lim_inf</a> <font color="Maroon" title="c5">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K6" title="KURATO_2:func.6">Lim_inf</a> <font color="Maroon" title="c6">D1</font>
 
<b>by </b><i><a class="txt" href="#E8:74"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="kurato_2.html#T52" target="_self" title="KURATO_2:th.52">Th54</a></i>;<br/>
<a NAME="E10:74"/><b>then </b><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="#E6:74"><i><font color="Green" title="E42">A4</font></i></a>, <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="c7">E1</font> being    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E12:74"/><i><font color="Green" title="E45">A7</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="c5">C1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E1</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <b>by </b><i><a class="txt" href="#E3:74"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="kurato_2.html#T67" target="_self" title="KURATO_2:th.67">Th69</a></i>;<br/>
<a NAME="E13:74"/>
 <a href="kurato_2.html#K6" title="KURATO_2:func.6">Lim_inf</a> <font color="Maroon" title="c5">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K6" title="KURATO_2:func.6">Lim_inf</a> <font color="Maroon" title="c7">E1</font>
 
<b>by </b><i><a class="txt" href="#E12:74"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="kurato_2.html#T52" target="_self" title="KURATO_2:th.52">Th54</a></i>;<br/>
<a NAME="E14:74"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="#E6:74"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="kurato_2.html#D12" target="_self" title="KURATO_2:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E15:74"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K7" title="KURATO_2:func.7">Lim_sup</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:74"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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>
