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<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   <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>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:53"/><i><font color="Green">E48</font></i>: 
( the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V3">idempotent</a> &amp; the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i><a class="ref" href="lattice2.html#T30">LATTICE2:30</a>, <a class="ref" href="lattice2.html#T31">LATTICE2:31</a>, <a class="ref" href="lattice2.html#T32">LATTICE2:32</a>, <a class="ref" href="lattice2.html#T73">LATTICE2:73</a></i>;<br/>
<a NAME="E2:53"/><i><font color="Green">E49</font></i>: 
( the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V3">idempotent</a> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i><a class="ref" href="lattice2.html#T26">LATTICE2:26</a>, <a class="ref" href="lattice2.html#T27">LATTICE2:27</a>, <a class="ref" href="lattice2.html#T29">LATTICE2:29</a>, <a class="ref" href="lattice2.html#T67">LATTICE2:67</a></i>;<br/>
<a NAME="E3:53"/><i><font color="Green">E50</font></i>: 
<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="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>thus </b><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="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#K7">`</a> 
<b>by </b><i><a class="ref" href="lattice4.html#D17" target="_self">Def17</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#K7">`</a> 
<b>by </b><i><a class="ref" href="lattice2.html#T68">LATTICE2:68</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="lattice4.html#T37" target="_self">Th37</a></i>
<br/>.= 
 <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="lattice2.html#T74">LATTICE2:74</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:53"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <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> holds <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="funct_2.html#K3">.</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="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="lattices.html#U1">L_meet</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="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <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>thus </b><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="funct_2.html#K3">.</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="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="lattices.html#K7">`</a> 
<b>by </b><i><a class="ref" href="lattice4.html#D17" target="_self">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <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>6</sub></font> <a href="lattices.html#K7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T51">LATTICES:51</a></i>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K7">`</a> </span>)</span>

<br/>.= 
the <a href="lattices.html#U1">L_meet</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="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D17" target="_self">Def17</a></i>
<br/>.= 
the <a href="lattices.html#U1">L_meet</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="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D17" target="_self">Def17</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattices.html#K7">`</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="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="lattices.html#K7">`</a> 
<b>by </b><i><a class="ref" href="lattice4.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<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>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D17" target="_self">Def17</a></i>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K5" target="_self">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E1:53"><i><font color="Green">E48</font></i></a>, <a class="txt" href="lattice4.html#E2:53"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice4.html#E3:53"><i><font color="Green">E50</font></i></a>, <a class="txt" href="lattice4.html#E4:53"><i><font color="Green">E51</font></i></a>, <a class="ref" href="setwiseo.html#T45">SETWISEO:45</a></i>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</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>
<b>by </b><i><a class="ref" href="tmap_1.html#T93">TMAP_1:93</a></i>
<br/>.= 
 <a href="lattice2.html#K3">FinMeet</a> <font color="Maroon">c<sub>2</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>
<b>by </b><i><a class="ref" href="lattice2.html#D4">LATTICE2:def 4</a></i>
;<br/>


</div>
