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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattices.html#V11">distributive</a> <a href="lattices.html#V14">upper-bounded</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>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="binop_1.html#NM1">UnOp</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>5</sub></font> = the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:34"/><i><font color="Green">E34</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> )
 
<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></i>;<br/>
<a NAME="E2:34"/><i><font color="Green">E35</font></i>: 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#R6">is_distributive_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#T35">LATTICE2:35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_1_1"/>
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">the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<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> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice4.html#E1:34"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice4.html#E2:34"><i><font color="Green">E35</font></i></a>, <a class="ref" href="setwiseo.html#T37">SETWISEO:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="binop_1.html#NM1">UnOp</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b>the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> = 
 <a href="lattice2.html#K3">FinMeet</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="lattice4.html#E1:34_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="lattice2.html#D4">LATTICE2:def 4</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="lattice4.html#E40"><i><font color="Green">Lemma26</font></i></a></i>
;<br/></div><b>end;</b></div><b>hence </b>the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</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"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>3</sub></font>

<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#T44">LATTICES:44</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">$$</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice4.html#E2:34_1_1_2"><i><font color="Green">E37</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:34"/>
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <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="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>2</sub></font>,<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> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>


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