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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:52_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> : ( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> is <a href="waybel_3.html#V1">compact</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E4:52_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">z'</font> is <a href="waybel_3.html#V1">compact</a> )
 ;<br/><a NAME="E5:52_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">z</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E6:52_1"/>
<font color="Maroon">z'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/><b>hence </b><a NAME="E7:52_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">x</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E2:52"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">x</font> : verum </span>}</span> 
 
<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="52_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">x</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">z'</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E3:52_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E4:52_2"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E5:52_2"/><b>then </b>
<font color="Maroon">z'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E7:52_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">z1</font> is <a href="waybel_3.html#V1">compact</a>
 <b>by </b><i/>;<br/><a NAME="E8:52_2"/>
<font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/><b>hence </b><a NAME="E9:52_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:52"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">x</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:52"/>
 <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">x</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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