<?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>assume </b><a NAME="E1:54"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">A1</font> is <a href="prob_1.html#V3">non-decreasing</a>
 ;<br/>

<a NAME="E2:54"/>
 <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:54_1_1_1"/>
<span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A1</font>
 <b>by </b><i>, <a class="ref" href="setlim_1.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E2:54_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T4" target="_self">Th4</a>, <a class="ref" href="prob_1.html#T29">PROB_1:29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:54_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A1</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:54_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A1</font>
 ;<br/><a NAME="E2:54_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="setlim_1.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E3:54_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_1.html#T29">PROB_1:29</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:54_1"/><b>then </b>
 <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:54_1"/>
 <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:54"/>
 <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1">Union</a> <font color="Maroon">A1</font>
 ;<br/>


</div>
