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<span class="kw">let </span><font color="Maroon" title="c1">BL</font> be   <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> holds  <span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Olive" title="b1">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></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> <font color="Maroon" title="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></span><br/>



<span class="kw">set </span><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/>
<span class="kw">set </span><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="E1:58"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
( 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> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  )
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<a NAME="E2:58"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
( 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> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  )
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<a NAME="E3:58"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
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#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</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="58_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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#K4" title="BINOP_1:func.4">.</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></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">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#K4" title="BINOP_1:func.4">.</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#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#D17" target="_self" title="LATTICE4:def.17">Def17</a></span>
<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#K3" title="LATTICES:func.3">"\/"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T50" title="LATTICES:th.50">LATTICES:50</a></span>
<br/>.= 
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#K4" title="BINOP_1:func.4">.</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#D17" target="_self" title="LATTICE4:def.17">Def17</a></span>
<br/>.= 
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#K4" title="BINOP_1:func.4">.</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#D17" target="_self" title="LATTICE4:def.17">Def17</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E44">A4</font></span>: <span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="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#U1" title="LATTICES:sel.1">L_meet</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#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#D17" target="_self" title="LATTICE4:def.17">Def17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T74" title="LATTICE2:th.74">LATTICE2:74</a></span>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T36" target="_self" title="LATTICE4:th.36">Th36</a></span>
<br/>.= 
 <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 class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T68" title="LATTICE2:th.68">LATTICE2:68</a></span>
;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</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#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">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#D4" title="LATTICE2:def.4">LATTICE2:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">BL</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">B</font>,<span class="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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#D17" target="_self" title="LATTICE4:def.17">Def17</a></span>
<br/>.= 
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">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="setwiseo.html#T45" title="SETWISEO:th.45">SETWISEO:45</a></span>
<br/>.= 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c2">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>
<br/>.= 
 <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#D3" title="LATTICE2:def.3">LATTICE2:def 3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
