<?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> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font>
 ;<br/><a NAME="E2:52_1"/><b>then </b>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span> : ( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Olive" title="b1">y</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c4">z'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span><b> such that </b><br/><a NAME="E4:52_1"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c4">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> &amp; <font color="Maroon" title="c2">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">z'</font> &amp; <font color="Maroon" title="c4">z'</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 ;<br/><a NAME="E5:52_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c3">z</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:52_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="waybel_8.html#T30" target="_self" title="WAYBEL_8:th.30">Th30</a></i>;<br/><a NAME="E6:52_1"/>
<font color="Maroon" title="c4">z'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:52_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/><b>hence </b><a NAME="E7:52_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">x</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_8.html#E4:52_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="waybel_8.html#E5:52_1"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:52"/><b>then </b><i><font color="Green" title="E26">A3</font></i>: 
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">x</font> : verum </span>}</span> 
 
<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="52_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">x</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c4">z'</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">x</font><b> such that </b><br/><a NAME="E3:52_2"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c4">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/><a NAME="E4:52_2"/>
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="waybel_8.html#T28" target="_self" title="WAYBEL_8:th.28">Th28</a></i>;<br/><a NAME="E5:52_2"/><b>then </b>
<font color="Maroon" title="c4">z'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">z1</font> = <font color="Maroon" title="c4">z'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_8.html#T28" target="_self" title="WAYBEL_8:th.28">Th28</a></i>;<br/><a NAME="E7:52_2"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c5">z1</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#T30" target="_self" title="WAYBEL_8:th.30">Th30</a></i>;<br/><a NAME="E8:52_2"/>
<font color="Maroon" title="c5">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/><b>hence </b><a NAME="E9:52_2"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="waybel_8.html#E3:52_2"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="waybel_8.html#E7:52_2"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:52"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">x</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font>
 
<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="E5:52"/>
 <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">x</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_8.html#E2:52"><i><font color="Green" title="E26">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>
