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

<b>let </b><font color="Maroon" title="c3">F</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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"> ( <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</font> iff <font color="Maroon" title="c3">F</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"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</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"> ( <font color="Maroon" title="c3">F</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"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span> implies <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:63_1_1_1"/><i><font color="Green" title="E47">B</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</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">F</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"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/><a NAME="E2:63_1_1_1"/>
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> <font color="Maroon" title="c3">F</font> iff <font color="Olive" title="b1">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"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> iff <font color="Maroon" title="c4">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"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</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"> ( <font color="Maroon" title="c4">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"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:63_1_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</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>  <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:63_1_1_1_1_1"/><i><font color="Green" title="E48">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>
 <b>by </b><i><a class="txt" href="#E1:63_1_1_1"><i><font color="Green" title="E47">B</font></i></a>, <a class="ref" href="kurato_2.html#T7" title="KURATO_2:th.7">KURATO_2:7</a></i>;<br/><a NAME="E4:63_1_1_1_1_1"/>
<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="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="#E3:63_1_1_1_1_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="setlim_1.html#T19" target="_self" title="SETLIM_1:th.19">Th21</a></i>;<br/><b>hence </b><a NAME="E5:63_1_1_1_1_1"/>
<font color="Maroon" title="c4">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"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><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>assume </b><a NAME="E2:63_1_1_1_1"/>
<font color="Maroon" title="c4">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"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:63_1_1_1_1"/><i><font color="Green" title="E48">B1</font></i>: 
<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="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/>
<a NAME="E5:63_1_1_1_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>
 
<b>by </b><i><a class="ref" href="setlim_1.html#T19" target="_self" title="SETLIM_1:th.19">Th21</a>, <a class="txt" href="#E4:63_1_1_1_1"><i><font color="Green" title="E48">B1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:63_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="#E1:63_1_1_1"><i><font color="Green" title="E47">B</font></i></a>, <a class="ref" href="kurato_2.html#T7" title="KURATO_2:th.7">KURATO_2:7</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="E3:63_1_1_1"/>
<font color="Maroon" title="c3">F</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"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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="E2:63_1_1"/><i><font color="Green" title="E47">B</font></i>: 
<font color="Maroon" title="c3">F</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"><a href="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</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">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</font></span><br/>

<a NAME="E3:63_1_1"/>
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> <font color="Maroon" title="c3">F</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</font> )</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"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:63_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</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>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</font></span><br/><b>then consider </b><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:63_1_1_2_1"/><i><font color="Green" title="E48">B1</font></i>: 
<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="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="#E2:63_1_1"><i><font color="Green" title="E47">B</font></i></a>, <a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></i>;<br/><a NAME="E4:63_1_1_2_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>
 <b>by </b><i><a class="ref" href="setlim_1.html#T19" target="_self" title="SETLIM_1:th.19">Th21</a>, <a class="txt" href="#E3:63_1_1_2_1"><i><font color="Green" title="E48">B1</font></i></a></i>;<br/><b>hence </b><a NAME="E5:63_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#T7" title="KURATO_2:th.7">KURATO_2:7</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="E2:63_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</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> <font color="Maroon" title="c3">F</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:63_1_1_2"/><i><font color="Green" title="E48">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>
 <b>by </b><i><a class="ref" href="kurato_2.html#T7" title="KURATO_2:th.7">KURATO_2:7</a></i>;<br/>
<a NAME="E5:63_1_1_2"/>
<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="setlim_1.html#K2" title="SETLIM_1:func.2">inferior_setsequence</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="#E4:63_1_1_2"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="setlim_1.html#T19" target="_self" title="SETLIM_1:th.19">Th21</a></i>;<br/>
<b>hence </b><a NAME="E6:63_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="#E2:63_1_1"><i><font color="Green" title="E47">B</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="E4:63_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="kurato_2.html#K3" title="KURATO_2:func.3">lim_inf</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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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