<?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>set </b><font color="Maroon">L</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</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/>
<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>;<br/>

<a NAME="E1:13"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font> is   <a href="waybel_0.html#NM4">Filter</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="E2:13"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Z</font> where <font color="Olive">Z</font> is   <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> : verum </span>}</span> 
 
;<br/>
<a NAME="E3:13"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel_0.html#D24">WAYBEL_0:def 24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:13_1"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/><b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:13"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E7:13"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>  st <font color="Olive">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:13_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">Y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> ;<br/><a NAME="E3:13_2_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:13_2"/><b>then </b>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<b>hence </b><a NAME="E4:13_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:13"/>
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/>


</div>
