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<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>assume </b><a NAME="E1:92_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">L</font> is <a href="lattices.html#V11">distributive</a> &amp; <font color="Maroon">L</font> is <a href="group_1.html#V6">finite</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as   <a href="lattice2.html#NM1">D0_Lattice</a> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">C</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E3:92_1_1"/><i><font color="Green">E18</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="lattice2.html#T30" target="_self">Th30</a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<a NAME="E4:92_1_1"/>
<font color="Maroon">L</font> is <a href="filter_0.html#V3">implicative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7">FILTER_0:def 7</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>] <b>means </b><font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span> ;<br/>
<a NAME="E1:92_1_1_1"/><i><font color="Green">E19</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S10">FRAENKEL:sch 10</a>();<br/></i>
<a NAME="E2:92_1_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="lattice2.html#T31" target="_self">Th31</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span>  as   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="lattice2.html#T32" target="_self">Th32</a>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>take </b><font color="Maroon">r</font> =  <a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span>;<br/>

<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:92_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:92_1_1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
 ex <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font> )
 ;<br/><span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T66">FUNCOP_1:66</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><b>hence </b><a NAME="E4:92_1_1_1_1"/>
<span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="lattice2.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:92_1_1_1"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice2.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E6:92_1_1_1"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="lattice2.html#T33" target="_self">Th33</a>, <a class="ref" href="lattice2.html#T21" target="_self">Th21</a></i>;<br/>

<b>let </b><font color="Maroon">r1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E7:92_1_1_1"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E8:92_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<a NAME="E9:92_1_1_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E10:92_1_1_1"/>
<font color="Maroon">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="lattice2.html#T34" target="_self">Th34</a>, <a class="ref" href="lattice2.html#T16" target="_self">Th16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:92_1_1"/>
<font color="Maroon">L</font> is <a href="lattice2.html#V1" target="_self">Heyting</a>
 <b>by </b><i><a class="ref" href="lattice2.html#T28" target="_self">Th28</a></i>;<br/>


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