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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattices.html#V17">Boolean</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="lattice4.html#M2" target="_self">ClosedSubset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:56"/><i><font color="Green">E50</font></i>: 
(  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:56"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">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" target="_self">FinJoin</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Olive">b<sub>1</sub></font> ) );<br/>
<a NAME="E3:56"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">{}.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="setwiseo.html#K2">.}</a></span>;<br/>

<i><font color="Green">E53</font></i>:  <a href="lattice2.html#K2">FinJoin</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="lattice4.html#E36"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
 <a href="lattice4.html#K2" target="_self">FinMeet</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="setwiseo.html#K2">.}</a></span>
<b>by </b><i><a class="ref" href="lattice4.html#T17" target="_self">Th17</a></i>
;<br/>
<a NAME="E3:56_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:56_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:56_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:56_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> = <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="setwiseo.html#K2">.}</a></span>;<br/>

<b>take </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="setwiseo.html#K2">.}</a></span>;<br/>

<b>thus </b><font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E2:56_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="lattices.html#T17">LATTICES:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#K7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T36" target="_self">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#K7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#K7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T48" target="_self">Th48</a></i>
;<br/>

<b>thus </b><a NAME="E2:56_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:56"><i><font color="Green">E50</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>

<b>thus </b><a NAME="E3:56_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:56"><i><font color="Green">E50</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:56_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:56_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2">FinJoin</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E2:56_1"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:56"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="setwiseo.html#K2">.}</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">c<sub>5</sub></font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:56_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">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" target="_self">FinJoin</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:56_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:56_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="setwiseo.html#T8">SETWISEO:8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:56_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E1:56_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="lattice4.html#E2:56_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="setwiseo.html#T8">SETWISEO:8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:56_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E3:56_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="lattice4.html#T44" target="_self">Th44</a></i>;<br/>
<i><font color="Green">E59</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K7">`</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E7:56_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="lattices.html#T51">LATTICES:51</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T49">LATTICES:49</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
;<br/>

<i><font color="Green">E60</font></i>:  <a href="lattice2.html#K2">FinJoin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E5:56_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="lattice4.html#E8:56_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="lattice4.html#T45" target="_self">Th45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T31">LATTICES:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T32" target="_self">Th32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice4.html#K2" target="_self">FinMeet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T32" target="_self">Th32</a></i>
<br/>.= 
 <a href="lattice4.html#K2" target="_self">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">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T30" target="_self">Th30</a></i>
;<br/>
<a NAME="E10:56_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b<sub>2</sub></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">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:56_2_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:56_2_3"/>
<span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T44">FUNCT_2:44</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="lattice4.html#E1:56_2_3"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:56_2_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E1:56_2_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<font color="Maroon">c<sub>13</sub></font> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="lattice4.html#E5:56_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funcop_1.html#T60">FUNCOP_1:60</a></i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>12</sub></font>

;<br/>
<b>hence </b><a NAME="E7:56_2_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E5:56_2_3"><i><font color="Green">E63</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:56_2"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:56_2_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:56_2_4"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E10:56_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E4:56_2_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice4.html#E5:56_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="lattice4.html#E3:56_2_4"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:56_2_4"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E7:56_2_4"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="setwiseo.html#K2">.}</a></span>;<br/>

<b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font>;<br/>

<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E3:56_2_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="lattice4.html#E6:56_2_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T21" target="_self">Th21</a></i>
;<br/>
<b>hence </b><a NAME="E2:56_2_4_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E7:56_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice4.html#E6:56_2_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="setwiseo.html#T8">SETWISEO:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:56_2_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:56_2_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:56_2_5"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="lattice4.html#E10:56_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E4:56_2_5"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice4.html#E5:56_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="lattice4.html#E3:56_2_5"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:56_2_5"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E7:56_2_5"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font>;<br/>

<b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="setwiseo.html#K2">.}</a></span>;<br/>

<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E3:56_2_5"><i><font color="Green">E63</font></i></a>, <a class="txt" href="lattice4.html#E6:56_2_5"><i><font color="Green">E64</font></i></a>, <a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#T45" target="_self">Th45</a></i>
;<br/>
<b>hence </b><a NAME="E2:56_2_5_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E7:56_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice4.html#E6:56_2_5"><i><font color="Green">E64</font></i></a>, <a class="ref" href="setwiseo.html#T8">SETWISEO:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:56_2_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2">FinJoin</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">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">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span></span>)</span> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="lattice4.html#E9:56_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="lattice4.html#E11:56_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:56"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4">SETWISEO:sch 4</a>(<a class="txt" href="lattice4.html#E3:56"><i><font color="Green">E52</font></i></a>, <a class="txt" href="lattice4.html#E4:56"><i><font color="Green">E53</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:56"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp;  <a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E2:56"><i><font color="Green">E51</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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:56_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">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" target="_self">FinJoin</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Olive">b<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> where B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:56_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:56_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>
<a NAME="E6:56_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice4.html#E1:56"><i><font color="Green">E50</font></i></a>, <a class="txt" href="lattice4.html#E3:56_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice4.html#E5:56_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="lattice4.html#T24" target="_self">Th24</a>, <a class="ref" href="lattice4.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E7:56_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E3:56_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice4.html#E5:56_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="lattice4.html#T48" target="_self">Th48</a></i>;<br/>
<b>hence </b><a NAME="E8:56_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lattice4.html#E6:56_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="lattice4.html#T44" target="_self">Th44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:56"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E7:56"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E10:56"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice4.html#K4" target="_self">SetImp</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="lattice2.html#K2">FinJoin</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2" target="_self">FinMeet</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="lattice4.html#E7:56"><i><font color="Green">E54</font></i></a></i>;<br/>


</div>
