<?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">Y</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span>;<br/>



<a NAME="E1:17"/><i><font color="Green">E33</font></i>: 
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> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">Y</font> is <a href="waybel_0.html#V2">filtered</a>
 ;<br/><a NAME="E2:17_1"/><b>then </b>
 <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#T22">WAYBEL_4:22</a></i>;<br/><a NAME="E3:17_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T19">YELLOW_1:19</a></i>;<br/><b>let </b><font color="Maroon">Z</font> be  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E4:17_1"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">Z</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">Z</font> as   <a href="struct_0.html#NM2">Subset</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="E7:17_1"/>
( ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies (  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">B</font> ) ) &amp; ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T43">WAYBEL_0:43</a>, <a class="ref" href="yellow_1.html#T20">YELLOW_1:20</a></i>;<br/><b>hence </b><a NAME="E8:17_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:17"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">Z</font> being <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>  st <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> holds <br/> <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">A</font> as  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font> as  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E4:17_2"/>
<font color="Maroon">A</font> <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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then </b> <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> = 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Z</font>
<b>by </b><i>, <a class="ref" href="yellow_1.html#T20">YELLOW_1:20</a></i>
<br/>.= 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Z</font>
<b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
;<br/><b>hence </b><a NAME="E6:17_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:17"/>
<font color="Maroon">Y</font> is <a href="waybel_0.html#V2">filtered</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T43">WAYBEL_0:43</a></i>;<br/>


</div>
