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

<b>let </b><font color="Maroon" title="c1">BL</font> be  <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">AF</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice4.html#M2" title="LATTICE4:mode.2">ClosedSubset</a> of <font color="Maroon" title="c1">BL</font>  st  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">AF</font> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">AF</font> holds <br/><span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b2">B</font></span>)</span> where <font color="Olive" title="b2">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Olive" title="b1">AF</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K3" title="LATTICE4:func.3">field_by</a> <font color="Olive" title="b1">AF</font></span><br/><b>let </b><font color="Maroon" title="c2">AF</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice4.html#M2" title="LATTICE4:mode.2">ClosedSubset</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> implies <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K3" title="LATTICE4:func.3">field_by</a> <font color="Maroon" title="c2">AF</font> )</span><br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E43">A1</font></i>: 
(  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K3" title="LATTICE4:func.3">field_by</a> <font color="Maroon" title="c2">AF</font></span><br/>

<b>set </b><font color="Maroon" title="c3">A1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span> ;<br/>
<a NAME="E2:55"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c2">AF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:55_1"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span> </span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">B</font> = <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span> as   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> ;<br/>
<font color="Maroon" title="c5">b</font> = 
<span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="ref" href="lattices.html#T39" title="LATTICES:th.39">LATTICES:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="ref" href="lattice4.html#T36" target="_self" title="LATTICE4:th.36">Th36</a></i>
;<br/>
<a NAME="E5:55_1"/><b>then </b>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font>
 
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="#E1:55_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="lattice4.html#T44" target="_self" title="LATTICE4:th.44">Th44</a></i>;<br/>
<a NAME="E6:55_1"/><b>then </b><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:55_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="ref" href="lattice4.html#T17" target="_self" title="LATTICE4:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E8:55_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E6:55_1"><i><font color="Green" title="E46">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:55"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> )</span><br/>

<b>assume </b><a NAME="E1:55_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">B</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E3:55_2"/><i><font color="Green" title="E45">A5</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c5">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> )
 ;<br/>
<b>thus </b><a NAME="E4:55_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>
 <b>by </b><i><a class="txt" href="#E3:55_2"><i><font color="Green" title="E45">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">A1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">BL</font> <b>by </b><i><a class="txt" href="#E2:55"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
<a NAME="E5:55"/><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c4">A1</font> is    <a href="lattice4.html#M3" title="LATTICE4:mode.3">Field</a> of <font color="Maroon" title="c1">BL</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice4.html#D14" target="_self" title="LATTICE4:def.14">LATTICE4:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>  st <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> holds <br/>( <font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> &amp; <font color="Maroon" title="c5">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> )</span><br/><b>let </b><font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> implies ( <font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> &amp; <font color="Maroon" title="c5">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> ) )</span><br/>



<b>assume </b><a NAME="E1:55_3"/><i><font color="Green" title="E46">A7</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> &amp; <font color="Maroon" title="c5">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> )</span><br/>

<b>thus </b><a NAME="E2:55_3"/>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c7">B1</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E2:55_3_1"/><i><font color="Green" title="E47">A8</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c7">B1</font> &amp; <font color="Maroon" title="c7">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> )
 <b>by </b><i><a class="txt" href="#E1:55_3"><i><font color="Green" title="E46">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">B2</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E4:55_3_1"/><i><font color="Green" title="E48">A9</font></i>: 
( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c8">B2</font> &amp; <font color="Maroon" title="c8">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> )
 <b>by </b><i><a class="txt" href="#E1:55_3"><i><font color="Green" title="E46">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E6:55_3_1"/><i><font color="Green" title="E49">A10</font></i>: 
<font color="Maroon" title="c9">B0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c8">B2</font>
 ;<br/>
<a NAME="E7:55_3_1"/><i><font color="Green" title="E50">A11</font></i>: 
<font color="Maroon" title="c9">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font>
 
<b>by </b><i><a class="txt" href="#E2:55_3_1"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="#E4:55_3_1"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="#E6:55_3_1"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E8:55_3_1"/>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c9">B0</font>
 
<b>by </b><i><a class="txt" href="#E2:55_3_1"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="#E4:55_3_1"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="#E6:55_3_1"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="lattice4.html#T30" target="_self" title="LATTICE4:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E9:55_3_1"/>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
 <b>by </b><i><a class="txt" href="#E7:55_3_1"><i><font color="Green" title="E50">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:55_3"/>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c7">B</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E2:55_3_2"/><i><font color="Green" title="E47">A12</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> )
 <b>by </b><i><a class="txt" href="#E1:55_3"><i><font color="Green" title="E46">A7</font></i></a></i>;<br/>
<a NAME="E3:55_3_2"/>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c7">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:55_3_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="lattice4.html#T48" target="_self" title="LATTICE4:th.48">Th48</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E5:55_3_2"/><i><font color="Green" title="E48">A13</font></i>: 
( <font color="Maroon" title="c8">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c5">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c8">B0</font> )
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="#E2:55_3_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="lattice4.html#T49" target="_self" title="LATTICE4:th.49">Th49</a></i>;<br/>
<b>thus </b><a NAME="E6:55_3_2"/>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
 <b>by </b><i><a class="txt" href="#E5:55_3_2"><i><font color="Green" title="E48">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">F</font> be    <a href="lattice4.html#M3" title="LATTICE4:mode.3">Field</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">AF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font> implies <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font> )</span><br/><b>assume </b><a NAME="E1:55_4"/><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c2">AF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font></span><br/><b>thus </b><a NAME="E2:55_4"/>
<font color="Maroon" title="c4">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> )</span><br/>

<b>assume </b><a NAME="E1:55_4_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">B</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E3:55_4_1"/><i><font color="Green" title="E47">A15</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> )
 ;<br/>
<a NAME="E4:55_4_1"/><i><font color="Green" title="E48">A16</font></i>: 
 <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> )</span><br/>

<b>assume </b><a NAME="E1:55_4_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E3:55_4_1_1"/><i><font color="Green" title="E49">A17</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> )
 ;<br/>
<b>thus </b><a NAME="E4:55_4_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="#E1:55_4"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E3:55_4_1_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="lattice4.html#T40" target="_self" title="LATTICE4:th.40">Th40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c8">F1</font> = <font color="Maroon" title="c5">F</font> as    <a href="lattice4.html#M2" title="LATTICE4:mode.2">ClosedSubset</a> of <font color="Maroon" title="c1">BL</font> <b>by </b><i><a class="ref" href="lattice4.html#T42" target="_self" title="LATTICE4:th.42">Th42</a></i>;<br/>
<a NAME="E6:55_4_1"/>
( <font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F1</font> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">F1</font> )
 
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="#E1:55_4"><i><font color="Green" title="E46">A14</font></i></a>, <a class="txt" href="#E3:55_4_1"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="#E4:55_4_1"><i><font color="Green" title="E48">A16</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E7:55_4_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="#E3:55_4_1"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="lattice4.html#T31" target="_self" title="LATTICE4:th.31">Th31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:55"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K3" title="LATTICE4:func.3">field_by</a> <font color="Maroon" title="c2">AF</font>
 <b>by </b><i><a class="txt" href="#E2:55"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="#E5:55"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="lattice4.html#D15" target="_self" title="LATTICE4:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
