<?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>set </b><font color="Maroon" title="c2">L</font> =  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>let </b><font color="Maroon" title="c3">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8" title="WAYBEL_0:func.8">Filt</a> <span class="p3">(<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></span>)</span></span>)</span>;<br/>

<div><i><font color="Green" title="E6">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/><a NAME="E2:12_1"/><b>then </b>
<font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8" title="WAYBEL_0:func.8">Filt</a> <span class="p3">(<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></span>)</span></span>)</span>
 ;<br/><a NAME="E3:12_1"/><b>then </b>
<font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="waybel_0.html#K8" title="WAYBEL_0:func.8">Filt</a> <span class="p2">(<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></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8" title="WAYBEL_0:func.8">Filt</a> <span class="p3">(<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></span>)</span></span>)</span> #)
 <b>by </b><i><a class="ref" href="yellow_1.html#D1" title="YELLOW_1:def.1">YELLOW_1:def 1</a></i>;<br/><a NAME="E4:12_1"/><b>then </b>
<font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D24" title="WAYBEL_0:def.24">WAYBEL_0:def 24</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">Z1</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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="E6:12_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c5">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z</font>
 ;<br/><b>thus </b><a NAME="E7:12_1"/>
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E6:12_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="waybel_4.html#T22" title="WAYBEL_4:th.22">WAYBEL_4:22</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:12"/>
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8" title="WAYBEL_0:func.8">Filt</a> <span class="p3">(<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></span>)</span></span>)</span>
 
;<br/>
<a NAME="E3:12"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="waybel_0.html#K8" title="WAYBEL_0:func.8">Filt</a> <span class="p2">(<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></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8" title="WAYBEL_0:func.8">Filt</a> <span class="p3">(<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></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1" title="YELLOW_1:def.1">YELLOW_1:def 1</a></i>;<br/>
<a NAME="E4:12"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<a NAME="E2:12_2"/><b>then </b>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K8" title="WAYBEL_0:func.8">Filt</a> <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="txt" href="waybel16.html#E3:12"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<a NAME="E3:12_2"/><b>then </b>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#D24" title="WAYBEL_0:def.24">WAYBEL_0:def 24</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">v1</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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="E5:12_2"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font>
 ;<br/>
<b>thus </b><a NAME="E6:12_2"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="txt" href="waybel16.html#E5:12_2"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:12"/><i><font color="Green" title="E9">A6</font></i>: 
for <font color="Olive" title="b1">Z</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>  st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> holds <br/><font color="Olive" title="b1">Z</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">Z</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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/>

<b>assume </b><a NAME="E1:12_3"/>
<font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/>

<a NAME="E2:12_3"/><b>then </b>
<font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K8" title="WAYBEL_0:func.8">Filt</a> <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="txt" href="waybel16.html#E3:12"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<a NAME="E3:12_3"/><b>then </b>
<font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#D24" title="WAYBEL_0:def.24">WAYBEL_0:def 24</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">Z1</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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="E5:12_3"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c5">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Z</font>
 ;<br/>
<b>thus </b><a NAME="E6:12_3"/>
<font color="Maroon" title="c4">Z</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 <b>by </b><i><a class="txt" href="waybel16.html#E5:12_3"><i><font color="Green" title="E10">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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/><b>assume </b><a NAME="E1:12_4"/>
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ;<br/><a NAME="E2:12_4"/><b>then </b>
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K8" title="WAYBEL_0:func.8">Filt</a> <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="txt" href="waybel16.html#E3:12"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/><a NAME="E3:12_4"/><b>then </b>
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">Z</font> where <font color="Olive" title="b1">Z</font> is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D24" title="WAYBEL_0:def.24">WAYBEL_0:def 24</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">V1</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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="E5:12_4"/><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c5">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">V</font>
 ;<br/><b>thus </b><a NAME="E6:12_4"/>
<font color="Maroon" title="c4">V</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 <b>by </b><i><a class="txt" href="waybel16.html#E5:12_4"><i><font color="Green" title="E10">A8</font></i></a></i>;<br/><b>thus </b><a NAME="E7:12_4"/>
<font color="Maroon" title="c4">V</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a>
 <b>by </b><i><a class="txt" href="waybel16.html#E5:12_4"><i><font color="Green" title="E10">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:12"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">Y</font> is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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="txt" href="waybel16.html#E1:12"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="waybel16.html#E4:12"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="waybel16.html#E5:12"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a>, <a class="ref" href="yellow_2.html#T39" title="YELLOW_2:th.39">YELLOW_2:39</a>, <a class="ref" href="yellow_2.html#T41" title="YELLOW_2:th.41">YELLOW_2:41</a></i>;<br/>


</div>
