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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">D</font> is <a href="lattice6.html#V9" target="_self">infimum-dense</a>
 ;<br/><a NAME="E2:51_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  ex <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a> </span>)</span> st <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Y</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">D'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E2:51_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">D'</font>,<font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E3:51_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">D'</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">D'</font> holds <br/><font color="Olive">b</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font> ) )
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<a NAME="E4:51_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_1"><b>proof </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="E1:51_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font>
 ;<br/>

<a NAME="E2:51_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
;<br/>
<b>hence </b><a NAME="E3:51_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D'</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:51_1_1"/>
for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> <a href="lattice6.html#K1" target="_self">%</a>  holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:51_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> <a href="lattice6.html#K1" target="_self">%</a> 
 ;<br/>

<a NAME="E2:51_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">d</font> <a href="lattice3.html#K4">%</a> </span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">u'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:51_1_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> <a href="lattice3.html#K4">%</a>  &amp; <font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> )
 ;<br/>
<a NAME="E5:51_1_1_2"/><i><font color="Green">E36</font></i>: 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">u'</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E9:51_1_1"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E7:51_1_1_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">u</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:51_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">D'</font> <a href="lattice6.html#K1" target="_self">%</a>  <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E8:51_1_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Maroon">D'</font> <a href="lattice6.html#K1" target="_self">%</a>  <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</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="51_1_1_3"><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="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:51_1_1_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">D'</font> <a href="lattice6.html#K1" target="_self">%</a>  <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font>
 ;<br/>

<i><font color="Green">E39</font></i>: <font color="Maroon">b</font> = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/>
<i><font color="Green">E40</font></i>: <font color="Maroon">a</font> = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/>
<a NAME="E4:51_1_1_3"/>
for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> holds <br/> <a href="lattice3.html#K5">%</a> <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Olive">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:51_1_1_3_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font>
 ;<br/>

<a NAME="E2:51_1_1_3_3"/><b>then </b>
<font color="Maroon">u</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">d</font> <a href="lattice3.html#K4">%</a> </span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 
;<br/>
<a NAME="E3:51_1_1_3_3"/><b>then </b>
<font color="Maroon">u</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> <a href="lattice6.html#K1" target="_self">%</a> 
 
;<br/>
<a NAME="E4:51_1_1_3_3"/><b>then </b>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">u</font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:51_1_1_3_3"/>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:51_1_1_3"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">D'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<a NAME="E6:51_1_1_3"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:51_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<b>hence </b><a NAME="E7:51_1_1_3"/>
<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="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:51_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">D'</font> <a href="lattice6.html#K1" target="_self">%</a> </span>)</span>,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E10:51_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> <a href="lattice6.html#K1" target="_self">%</a>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_4"><b>proof </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="E1:51_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> <a href="lattice6.html#K1" target="_self">%</a> 
 ;<br/>

<a NAME="E2:51_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">d</font> <a href="lattice3.html#K4">%</a> </span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> </span>}</span> 
 
;<br/>
<a NAME="E3:51_1_1_4"/><b>then </b>
 ex <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font> <a href="lattice3.html#K4">%</a>  &amp; <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> )
 
;<br/>
<a NAME="E4:51_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">d</font> <a href="lattice3.html#K4">%</a> </span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E5:51_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:51_1_1"/><b>then </b>
<font color="Maroon">D'</font> <a href="lattice6.html#K1" target="_self">%</a>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:51_1_1"/>
 ex <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a> </span>)</span> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Y</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:51_1"/>
<font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a>  is <a href="waybel_6.html#V4">order-generating</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#T15">WAYBEL_6:15</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a>  is <a href="waybel_6.html#V4">order-generating</a>
 ;<br/><a NAME="E2:51_2"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">D'</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> st <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">D'</font>,<font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>consider </b><font color="Maroon">Y</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a> </span>)</span><b> such that </b><br/><a NAME="E2:51_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_6.html#T15">WAYBEL_6:15</a></i>;<br/>
<a NAME="E3:51_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></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="Maroon">a</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font> ) )
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E4:51_2_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">x</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="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_1"><b>proof </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="E1:51_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:51_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a> 
 
;<br/>
<b>hence </b><a NAME="E3:51_2_1_1"/>
<font color="Maroon">x</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="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:51_2_1"/>
for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="lattice6.html#K2" target="_self">%</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:51_2_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="lattice6.html#K2" target="_self">%</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:51_2_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">q'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E4:51_2_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>
<a NAME="E5:51_2_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q'</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<font color="Maroon">q'</font> = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">q'</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">q'</font></span>)</span> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E7:51_2_1_2"/>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:51_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#R3">is_less_than</a>  <a href="lattice6.html#K2" target="_self">%</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<a NAME="E8:51_2_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="lattice3.html#R3">is_less_than</a>  <a href="lattice6.html#K2" target="_self">%</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_3"><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 <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:51_2_1_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#R3">is_less_than</a>  <a href="lattice6.html#K2" target="_self">%</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:51_2_1_3"/>
for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:51_2_1_3_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:51_2_1_3_1"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> </span>}</span> 
 
;<br/>
<a NAME="E3:51_2_1_3_1"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="lattice6.html#K2" target="_self">%</a> <font color="Maroon">Y</font>
 
;<br/>
<a NAME="E4:51_2_1_3_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">u</font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E6:51_2_1_3_1"/>
<font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E1:51_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:51_2_1_3"/><b>then </b>
<font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  <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="E4:51_2_1_3"/><b>then </b>
<font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<b>hence </b><a NAME="E5:51_2_1_3"/>
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:51_2_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <span class="p1">(<span class="default"><a href="lattice6.html#K2" target="_self">%</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<a NAME="E10:51_2_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="lattice6.html#K2" target="_self">%</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_4"><b>proof </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="E1:51_2_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lattice6.html#K2" target="_self">%</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:51_2_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E4:51_2_1_4"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 ;<br/>
<a NAME="E5:51_2_1_4"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:51_2"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E6:51_2_1_4"/><b>then </b>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">d</font> <a href="lattice3.html#K4">%</a> </span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:51_2_1_4"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="lattice3.html#K4">%</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 ;<br/>
<font color="Maroon">x</font> = 
<font color="Maroon">y</font> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="txt" href="lattice6.html#E1:51_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="lattice6.html#E2:51_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E10:51_2_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E2:51_2_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:51_2_1"/><b>then </b>
 <a href="lattice6.html#K2" target="_self">%</a> <font color="Maroon">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:51_2_1"/>
 ex <font color="Olive">D'</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Olive">D'</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:51_2"/>
<font color="Maroon">D</font> is <a href="lattice6.html#V9" target="_self">infimum-dense</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:51"/>
( <font color="Maroon">D</font> is <a href="lattice6.html#V9" target="_self">infimum-dense</a> iff <font color="Maroon">D</font> <a href="lattice6.html#K1" target="_self">%</a>  is <a href="waybel_6.html#V4">order-generating</a> )
 <b>by </b><i/>;<br/>


</div>
