<?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>;<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>;<br/>



<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E50">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> )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">B</font> be   <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/>

<b>assume </b><a NAME="E2:56"/><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon" title="c3">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>set </b><font color="Maroon" title="c4">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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>means </b>( $1 <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> implies  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b2">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b3">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b2">A1</font>, <font color="Olive" title="b3">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> $1,<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B0</font> ) );<br/>
<a NAME="E3:56"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</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="56_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1"/>
 <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</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>take </b><font color="Maroon" title="c5">B0</font> = <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>;<br/>

<i><font color="Green" title="E53">A4</font></i>:  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font></span>)</span>,<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> = 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font>
<b>by </b><i><a class="txt" href="lattice4.html#E38"><i><font color="Green" title="E22">Lm3</font></i></a></i>
<br/>.= 
 <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>
<b>by </b><i><a class="ref" href="lattice4.html#T17" target="_self" title="LATTICE4:th.17">Th17</a></i>
;<br/>
<a NAME="E3:56_1"/>
<font color="Maroon" title="c5">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2"><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><br/>

<b>assume </b><a NAME="E1:56_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font>
 ;<br/>

<a NAME="E2:56_1_2"/><b>then </b><i><font color="Green" title="E54">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:56_1_2"/>
 ex <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c7">A1</font> = <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>;<br/>

<b>take </b><font color="Maroon" title="c8">A2</font> = <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><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></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>;<br/>

<b>thus </b><font color="Maroon" title="c6">x</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> <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>
<b>by </b><i><a class="txt" href="lattice4.html#E2:56_1_2"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="lattices.html#T17" title="LATTICES:th.17">LATTICES:17</a></i>
<br/>.= 
<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> <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>
<b>by </b><i><a class="ref" href="lattice4.html#T36" target="_self" title="LATTICE4:th.36">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c7">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <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>
<b>by </b><i><a class="ref" href="lattice4.html#T16" target="_self" title="LATTICE4:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c7">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</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="Maroon" title="c8">A2</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T17" target="_self" title="LATTICE4:th.17">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c7">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c8">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T48" target="_self" title="LATTICE4:th.48">Th48</a></i>
;<br/>

<b>thus </b><a NAME="E2:56_1_2_1"/>
<font color="Maroon" title="c7">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:56"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>

<b>thus </b><a NAME="E3:56_1_2_1"/>
<font color="Maroon" title="c8">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:56"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:56_1_2"/>
<font color="Maroon" title="c6">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"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:56_1"/>
( <font color="Maroon" title="c5">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font></span>)</span>,<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> <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">B0</font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E2:56_1"><i><font color="Green" title="E53">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:56"/><i><font color="Green" title="E53">A6</font></i>: 
for <font color="Olive" title="b1">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><br/> for <font color="Olive" title="b2">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">S<sub>1</sub></font>[<font color="Olive" title="b1">B'</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B'</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">B'</font> be   <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/><b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>;<br/>



<b>assume </b><a NAME="E1:56_2"/><i><font color="Green" title="E54">A7</font></i>: 
( <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> implies  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b2">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b3">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b2">A1</font>, <font color="Olive" title="b3">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c5">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B1</font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:56_2"/><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon" title="c5">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="E3:56_2"/><b>then </b><i><font color="Green" title="E56">A9</font></i>: 
( <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> &amp; <font color="Maroon" title="c6">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="ref" href="setwiseo.html#T8" title="SETWISEO:th.8">SETWISEO:8</a></i>;<br/>
<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="E5:56_2"/><i><font color="Green" title="E57">A10</font></i>: 
( <font color="Maroon" title="c7">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c5">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> <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> )
 <b>by </b><i><a class="txt" href="lattice4.html#E1:56_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="lattice4.html#E2:56_2"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="setwiseo.html#T8" title="SETWISEO:th.8">SETWISEO:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">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="E7:56_2"/><i><font color="Green" title="E58">A11</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <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="c9">q</font> &amp; <font color="Maroon" title="c8">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="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E3:56_2"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="lattice4.html#T44" target="_self" title="LATTICE4:th.44">Th44</a></i>;<br/>
<i><font color="Green" title="E59">A12</font></i>: <font color="Maroon" title="c6">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E7:56_2"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="lattices.html#T51" title="LATTICES:th.51">LATTICES:51</a></i>
<br/>.= 
<font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T49" title="LATTICES:th.49">LATTICES:49</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c10">J</font> = the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c8">p</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span>
;<br/>

<i><font color="Green" title="E60">A13</font></i>:  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B'</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<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> = 
<span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c7">B1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E5:56_2"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="lattice4.html#E8:56_2"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="lattice4.html#T45" target="_self" title="LATTICE4:th.45">Th45</a></i>
<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="Maroon" title="c7">B1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</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="Maroon" title="c7">B1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T31" title="LATTICES:th.31">LATTICES:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c8">p</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</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="Maroon" title="c7">B1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T32" target="_self" title="LATTICE4:th.32">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c8">p</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T32" target="_self" title="LATTICE4:th.32">Th32</a></i>
<br/>.= 
 <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c8">p</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T30" target="_self" title="LATTICE4:th.30">Th30</a></i>
;<br/>
<a NAME="E10:56_2"/><i><font color="Green" title="E61">A14</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">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="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Olive" title="b2">b</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font> holds <br/> ex <font color="Olive" title="b3">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B1</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c12">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>;<br/>



<b>assume </b><a NAME="E1:56_2_3"/><i><font color="Green" title="E62">A15</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c12">b</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font>
 ;<br/>

<a NAME="E2:56_2_3"/>
<span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c12">b</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font> <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>
 
<b>by </b><i><a class="ref" href="funct_2.html#T44" title="FUNCT_2:th.44">FUNCT_2:44</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c11">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> <b>by </b><i><a class="txt" href="lattice4.html#E1:56_2_3"><i><font color="Green" title="E62">A15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">a</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="E5:56_2_3"/><i><font color="Green" title="E63">A16</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B1</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c12">b</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E1:56_2_3"><i><font color="Green" title="E62">A15</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<font color="Maroon" title="c13">x</font> = 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<font color="Maroon" title="c12">b</font>
<b>by </b><i><a class="txt" href="lattice4.html#E5:56_2_3"><i><font color="Green" title="E63">A16</font></i></a>, <a class="ref" href="funcop_1.html#T60" title="FUNCOP_1:th.60">FUNCOP_1:60</a></i>
<br/>.= 
<font color="Maroon" title="c14">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">b</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T91" title="TMAP_1:th.91">TMAP_1:91</a></i>
;<br/>
<b>hence </b><a NAME="E7:56_2_3"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B1</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">b</font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E5:56_2_3"><i><font color="Green" title="E63">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:56_2"/><i><font color="Green" title="E62">A17</font></i>: 
<span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c8">p</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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><br/>

<b>assume </b><a NAME="E1:56_2_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c8">p</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">a</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="E3:56_2_4"/><i><font color="Green" title="E63">A18</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B1</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">p</font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E10:56_2"><i><font color="Green" title="E61">A14</font></i></a></i>;<br/>
<a NAME="E4:56_2_4"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice4.html#E5:56_2"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="lattice4.html#E3:56_2_4"><i><font color="Green" title="E63">A18</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">A1</font>, <font color="Maroon" title="c14">A2</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="E6:56_2_4"/><i><font color="Green" title="E64">A19</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c14">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 ;<br/>
<a NAME="E7:56_2_4"/>
 ex <font color="Olive" title="b1">A1'</font>, <font color="Olive" title="b2">A2'</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> st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1'</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2'</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">A1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_4_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c15">A1'</font> = <font color="Maroon" title="c13">A1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c8">p</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>;<br/>

<b>take </b><font color="Maroon" title="c16">A2'</font> = <font color="Maroon" title="c14">A2</font>;<br/>

<font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c14">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E3:56_2_4"><i><font color="Green" title="E63">A18</font></i></a>, <a class="txt" href="lattice4.html#E6:56_2_4"><i><font color="Green" title="E64">A19</font></i></a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A1'</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c16">A2'</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T21" target="_self" title="LATTICE4:th.21">Th21</a></i>
;<br/>
<b>hence </b><a NAME="E2:56_2_4_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A1'</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c16">A2'</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c15">A1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c16">A2'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E7:56_2"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="lattice4.html#E6:56_2_4"><i><font color="Green" title="E64">A19</font></i></a>, <a class="ref" href="setwiseo.html#T8" title="SETWISEO:th.8">SETWISEO:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:56_2_4"/>
<font color="Maroon" title="c11">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"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:56_2"/>
<span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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><br/>

<b>assume </b><a NAME="E1:56_2_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">a</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="E3:56_2_5"/><i><font color="Green" title="E63">A20</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B1</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="lattice4.html#E10:56_2"><i><font color="Green" title="E61">A14</font></i></a></i>;<br/>
<a NAME="E4:56_2_5"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice4.html#E5:56_2"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="lattice4.html#E3:56_2_5"><i><font color="Green" title="E63">A20</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">A1</font>, <font color="Maroon" title="c14">A2</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="E6:56_2_5"/><i><font color="Green" title="E64">A21</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c14">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 ;<br/>
<a NAME="E7:56_2_5"/>
 ex <font color="Olive" title="b1">A1'</font>, <font color="Olive" title="b2">A2'</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> st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1'</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2'</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">A1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_5_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c15">A1'</font> = <font color="Maroon" title="c13">A1</font>;<br/>

<b>take </b><font color="Maroon" title="c16">A2'</font> = <font color="Maroon" title="c14">A2</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">q</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>;<br/>

<font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c14">A2</font>,<span class="p3">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E3:56_2_5"><i><font color="Green" title="E63">A20</font></i></a>, <a class="txt" href="lattice4.html#E6:56_2_5"><i><font color="Green" title="E64">A21</font></i></a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A1'</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c16">A2'</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T45" target="_self" title="LATTICE4:th.45">Th45</a></i>
;<br/>
<b>hence </b><a NAME="E2:56_2_5_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A1'</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c16">A2'</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c15">A1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c16">A2'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E7:56_2"><i><font color="Green" title="E58">A11</font></i></a>, <a class="txt" href="lattice4.html#E6:56_2_5"><i><font color="Green" title="E64">A21</font></i></a>, <a class="ref" href="setwiseo.html#T8" title="SETWISEO:th.8">SETWISEO:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:56_2_5"/>
<font color="Maroon" title="c11">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"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:56_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c8">p</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B'</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c8">p</font></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c7">B1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="lattice4.html#E9:56_2"><i><font color="Green" title="E60">A13</font></i></a>, <a class="txt" href="lattice4.html#E11:56_2"><i><font color="Green" title="E62">A17</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:56"/>
for <font color="Olive" title="b1">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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4" title="SETWISEO:sch.4">SETWISEO:sch 4</a>(<a class="txt" href="lattice4.html#E3:56"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="lattice4.html#E4:56"><i><font color="Green" title="E53">A6</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c5">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="E7:56"/><i><font color="Green" title="E54">A22</font></i>: 
( <font color="Maroon" title="c5">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c3">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> <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">B1</font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E2:56"><i><font color="Green" title="E51">A2</font></i></a></i>;<br/>
<a NAME="E8:56"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span>  <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3"><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><br/>

<b>assume </b><a NAME="E1:56_3"/>
<font color="Maroon" title="c6">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"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</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">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">A1</font>, <font color="Maroon" title="c8">A2</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="E3:56_3"/><i><font color="Green" title="E55">A23</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c7">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c8">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c8">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 ;<br/>
<b>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="E5:56_3"/><i><font color="Green" title="E56">A24</font></i>: 
( <font color="Maroon" title="c9">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="c8">A2</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c7">A1</font> )
 ;<br/>
<a NAME="E6:56_3"/><i><font color="Green" title="E57">A25</font></i>: 
( <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> )
 
<b>by </b><i><a class="txt" href="lattice4.html#E1:56"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="lattice4.html#E3:56_3"><i><font color="Green" title="E55">A23</font></i></a>, <a class="txt" href="lattice4.html#E5:56_3"><i><font color="Green" title="E56">A24</font></i></a>, <a class="ref" href="lattice4.html#T24" target="_self" title="LATTICE4:th.24">Th24</a>, <a class="ref" href="lattice4.html#T31" target="_self" title="LATTICE4:th.31">Th31</a></i>;<br/>
<a NAME="E7:56_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <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="c10">q</font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E3:56_3"><i><font color="Green" title="E55">A23</font></i></a>, <a class="txt" href="lattice4.html#E5:56_3"><i><font color="Green" title="E56">A24</font></i></a>, <a class="ref" href="lattice4.html#T48" target="_self" title="LATTICE4:th.48">Th48</a></i>;<br/>
<b>hence </b><a NAME="E8:56_3"/>
<font color="Maroon" title="c6">x</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="lattice4.html#E6:56_3"><i><font color="Green" title="E57">A25</font></i></a>, <a class="ref" href="lattice4.html#T44" target="_self" title="LATTICE4:th.44">Th44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:56"/><b>then </b>
<font color="Maroon" title="c5">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="lattice4.html#E7:56"><i><font color="Green" title="E54">A22</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="E10:56"/>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">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;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="c3">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B0</font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E7:56"><i><font color="Green" title="E54">A22</font></i></a></i>;<br/>


</div>
