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<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">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>set </b><font color="Maroon" title="c3">M</font> = the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font>;<br/>
<b>set </b><font color="Maroon" title="c4">J</font> = the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font>;<br/>
<a NAME="E1:53"/><i><font color="Green" title="E44">A1</font></i>: 
( the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  )
 
<b>by </b><i><a class="ref" href="lattice2.html#T30" title="LATTICE2:th.30">LATTICE2:30</a>, <a class="ref" href="lattice2.html#T31" title="LATTICE2:th.31">LATTICE2:31</a>, <a class="ref" href="lattice2.html#T32" title="LATTICE2:th.32">LATTICE2:32</a>, <a class="ref" href="lattice2.html#T73" title="LATTICE2:th.73">LATTICE2:73</a></i>;<br/>
<a NAME="E2:53"/><i><font color="Green" title="E45">A2</font></i>: 
( the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp; the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  )
 
<b>by </b><i><a class="ref" href="lattice2.html#T26" title="LATTICE2:th.26">LATTICE2:26</a>, <a class="ref" href="lattice2.html#T27" title="LATTICE2:th.27">LATTICE2:27</a>, <a class="ref" href="lattice2.html#T29" title="LATTICE2:th.29">LATTICE2:29</a>, <a class="ref" href="lattice2.html#T67" title="LATTICE2:th.67">LATTICE2:67</a></i>;<br/>
<a NAME="E3:53"/><i><font color="Green" title="E46">A3</font></i>: 
<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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</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" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
<b>by </b><i><a class="ref" href="lattice4.html#D17" target="_self" title="LATTICE4:def.17">Def17</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#K7" title="LATTICES:func.7">`</a> 
<b>by </b><i><a class="ref" href="lattice2.html#T68" title="LATTICE2:th.68">LATTICE2:68</a></i>
<br/>.= 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font>
<b>by </b><i><a class="ref" href="lattice4.html#T37" target="_self" title="LATTICE4:th.37">Th37</a></i>
<br/>.= 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font>
<b>by </b><i><a class="ref" href="lattice2.html#T74" title="LATTICE2:th.74">LATTICE2:74</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:53"/><i><font color="Green" title="E47">A4</font></i>: 
for <font color="Olive" title="b1">a</font>, <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> holds  <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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</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="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</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" title="c5">a</font>, <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>thus </b><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="funct_2.html#K3" title="FUNCT_2:func.3">.</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="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
<b>by </b><i><a class="ref" href="lattice4.html#D17" target="_self" title="LATTICE4:def.17">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <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="c6">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T51" title="LATTICES:th.51">LATTICES:51</a></i>
<br/>.= 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</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="lattice4.html#K5" title="LATTICE4:func.5">comp</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="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D17" target="_self" title="LATTICE4:def.17">Def17</a></i>
<br/>.= 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</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="lattice4.html#K5" title="LATTICE4:func.5">comp</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="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><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> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D17" target="_self" title="LATTICE4:def.17">Def17</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
<b>by </b><i><a class="ref" href="lattice2.html#D3" title="LATTICE2:def.3">LATTICE2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice4.html#D17" target="_self" title="LATTICE4:def.17">Def17</a></i>
<br/>.= 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E1:53"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="lattice4.html#E2:53"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="lattice4.html#E3:53"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="lattice4.html#E4:53"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="setwiseo.html#T45" title="SETWISEO:th.45">SETWISEO:45</a></i>
<br/>.= 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T93" title="TMAP_1:th.93">TMAP_1:93</a></i>
<br/>.= 
 <a href="lattice2.html#K3" title="LATTICE2:func.3">FinMeet</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice2.html#D4" title="LATTICE2:def.4">LATTICE2:def 4</a></i>
;<br/>


</div>
