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<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">F</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:53"/>
( <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span> implies <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:53_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</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><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:53_1"/>
( <font color="Maroon">F'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">F'</font> &amp; <font color="Maroon">F'</font> is <a href="waybel_3.html#V1">compact</a> )
 
<b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E5:53_1"/><b>then </b>
<font color="Maroon">F'</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">F'</font>
 
;<br/>
<a NAME="E6:53_1"/><b>then </b>
<font color="Maroon">F'</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">F'</font> : verum </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">F1</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F'</font><b> such that </b><br/><a NAME="E8:53_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font>
 ;<br/>
<b>thus </b><a NAME="E9:53_1"/>
<font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:53"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</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="E4:53"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
;<br/>
<a NAME="E5:53"/><b>then </b>
<font color="Maroon">F</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">F'</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_8.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:53"/><b>then </b>
<font color="Maroon">F'</font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2" target="_self">compactbelow</a> <font color="Maroon">F'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:53"/><b>then </b>
<font color="Maroon">F'</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:53"/>
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" target="_self">CompactSublatt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div>
