<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V2">with_infima</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>reconsider </b><font color="Maroon">G1</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="txt" href="waybel12.html#E1"><i><font color="Green">Lemma18</font></i></a></i>;<br/>
<a NAME="E2:38"/>
<font color="Maroon">G1</font> is <a href="waybel_0.html#V2">filtered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D2">WAYBEL_0:def 2</a><br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E4:38_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 ;<br/>
<b>assume </b><a NAME="E5:38_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:38_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>and </b><br/><a NAME="E8:38_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y1</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 ;<br/>
<b>take </b><font color="Maroon">z</font> = <font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y1</font>;<br/>

<a NAME="E9:38_1"/>
<font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:38_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:38_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:38_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E5:38_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:38_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E6:38_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>
<a NAME="E7:38_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel12.html#E4:38_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>;<br/>
<a NAME="E8:38_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y1</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">t</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>
<a NAME="E9:38_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y1</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:38_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="txt" href="waybel12.html#E7:38_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel12.html#E8:38_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:38_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:38_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 ;<br/>

<b>thus </b><a NAME="E11:38_1"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> </span>}</span>  is  <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 ;<br/>


</div>
