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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A1</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon" title="c2">A1</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a>
 ;<br/>

<a NAME="E2:64"/>
 <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c2">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:64_1_1"/>
<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"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c2">A1</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E3:64_1_1"/><i><font color="Green" title="E52">A2</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"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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>thus </b><a NAME="E4:64_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c2">A1</font>
 <b>by </b><i><a class="txt" href="setlim_1.html#E1:64"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="setlim_1.html#E3:64_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="setlim_1.html#T58" target="_self" title="SETLIM_1:th.58">Th58</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:64_1"/><b>then </b><i><font color="Green" title="E52">A3</font></i>: 
 <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c2">A1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:64_1_2"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c2">A1</font>
 ;<br/><a NAME="E2:64_1_2"/><b>then </b>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0
 <b>by </b><i><a class="ref" href="setlim_1.html#T22" target="_self" title="SETLIM_1:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E3:64_1_2"/>
<font color="Maroon" title="c3">y</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#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c2">A1</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>;<br/></div><b>end;</b></div>
<a NAME="E4:64_1"/><b>then </b>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c2">A1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:64_1"/>
 <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c2">A1</font>
 <b>by </b><i><a class="txt" href="setlim_1.html#E2:64_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:64"/>
 <a href="kurato_2.html#K1" title="KURATO_2:func.1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" title="SETLIM_1:func.1">inferior_setsequence</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c2">A1</font>
 ;<br/>


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