<?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="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">D</font> is <a href="lattice6.html#V8" target="_self">supremum-dense</a>
 ;<br/><b>thus </b><a NAME="E2:49_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) </span>}</span> ,<font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_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 <font color="Maroon">L</font>;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</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="E2:49_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">D'</font>,<font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E3:49_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> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_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="E1:49_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font>
 ;<br/>

<a NAME="E2:49_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>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E4:49_1_1_1"/>
<font color="Maroon">D'</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E5:49_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E6:49_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:49_1_1"/><b>then </b>
<font color="Maroon">D'</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:49_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T46">LATTICE3:46</a></i>;<br/>
<a NAME="E6:49_1_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> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span>  holds <br/><font color="Olive">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_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="E1:49_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:49_1_1_2"/><b>then </b>
 ex <font color="Olive">q'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Olive">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Olive">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> )
 
;<br/>
<b>hence </b><a NAME="E3:49_1_1_2"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:49_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<a NAME="E8:49_1_1"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E9:49_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) </span>}</span> ,<font color="Maroon">L</font>
 ;<br/><a NAME="E2:49_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#K15">"\/"</a> <font color="Olive">D'</font>,<font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_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>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span> ;<br/>
<a NAME="E1:49_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> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span>  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="49_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="E1:49_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:49_2_1_1"/><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="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Olive">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> )
 
;<br/>
<b>hence </b><a NAME="E3:49_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:49_2_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span>  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/>
<a NAME="E3:49_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:49_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#K15">"\/"</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="E3:49_2"/>
<font color="Maroon">D</font> is <a href="lattice6.html#V8" target="_self">supremum-dense</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:49"/>
( <font color="Maroon">D</font> is <a href="lattice6.html#V8" target="_self">supremum-dense</a> iff for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> 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> &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) </span>}</span> ,<font color="Maroon">L</font> )
 <b>by </b><i/>;<br/>


</div>
