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

<b>let </b><font color="Maroon">BL</font> be  <a href="lattices.html#V17">Boolean</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">AF</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="lattice4.html#M2" target="_self">ClosedSubset</a> of <font color="Maroon">BL</font>;<br/>



<b>assume </b><a NAME="E1:57"/><i><font color="Green">E25</font></i>: 
(  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">BL</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AF</font> &amp;  <a href="lattices.html#K6">Top</a> <font color="Maroon">BL</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AF</font> )
 ;<br/>

<b>set </b><font color="Maroon">A1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font> : <font color="Olive">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> </span>}</span> ;<br/>
<a NAME="E2:57"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">AF</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font> : <font color="Olive">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:57_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AF</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font> ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">b</font></span><a href="setwiseo.html#K2">.}</a></span> as   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font> ;<br/>
<font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">BL</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattices.html#T39">LATTICES:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">BL</font></span>)</span> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font>
<b>by </b><i/>
;<br/>
<a NAME="E5:57_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font>
 
<b>by </b><i>, , <a class="ref" href="lattice4.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E6:57_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:57_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:57_1"/>
<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="lattice4.html#K2" target="_self">FinMeet</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font> : <font color="Olive">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:57"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font> : <font color="Olive">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:57_2"/>
<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="lattice4.html#K2" target="_self">FinMeet</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font> : <font color="Olive">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E3:57_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> )
 ;<br/>
<b>thus </b><a NAME="E4:57_2"/>
<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">BL</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font> : <font color="Olive">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">BL</font> <b>by </b><i>, <a class="ref" href="lattice4.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:57"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">A1</font> is    <a href="lattice4.html#M3" target="_self">Field</a> of <font color="Maroon">BL</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice4.html#D14" target="_self">LATTICE4:def 14</a><br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font>;<br/>



<b>assume </b><a NAME="E1:57_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 ;<br/>

<b>thus </b><a NAME="E2:57_3"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">B1</font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E2:57_3_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B2</font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E4:57_3_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B0</font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E6:57_3_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">B0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">B2</font>
 ;<br/>
<a NAME="E7:57_3_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">B0</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font>
 
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E8:57_3_1"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">B0</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E9:57_3_1"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:57_3"/>
<font color="Maroon">p</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">B</font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E2:57_3_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> )
 <b>by </b><i/>;<br/>
<a NAME="E3:57_3_2"/>
<font color="Maroon">p</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a>  <a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">BL</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lattice4.html#T8" target="_self">Th8</a></i>;<br/>
<b>then consider </b><font color="Maroon">B0</font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E5:57_3_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">B0</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">B0</font> )
 <b>by </b><i>, , <a class="ref" href="lattice4.html#T9" target="_self">Th9</a></i>;<br/>
<b>thus </b><a NAME="E6:57_3_2"/>
<font color="Maroon">p</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">F</font> be    <a href="lattice4.html#M3" target="_self">Field</a> of <font color="Maroon">BL</font>;<br/><b>assume </b><a NAME="E1:57_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">AF</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 ;<br/><b>thus </b><a NAME="E2:57_4"/>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:57_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E3:57_4_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> )
 ;<br/>
<a NAME="E4:57_4_1"/><i><font color="Green">E70</font></i>: 
 <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:57_4_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E3:57_4_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AF</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AF</font> )
 ;<br/>
<b>thus </b><a NAME="E4:57_4_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="lattice4.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as    <a href="lattice4.html#M2" target="_self">ClosedSubset</a> of <font color="Maroon">BL</font> <b>by </b><i/>;<br/>
<a NAME="E6:57_4_1"/>
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F1</font> &amp;  <a href="lattices.html#K6">Top</a> <font color="Maroon">BL</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> )
 
<b>by </b><i>, , , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E7:57_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:57"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">BL</font> : <font color="Olive">B</font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">AF</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K3" target="_self">field_by</a> <font color="Maroon">AF</font>
 <b>by </b><i><a class="ref" href="lattice4.html#D4" target="_self">Def4</a>, , </i>;<br/>


</div>
