<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K7">[:]</a> <span class="p1">(<span class="default"><a href="heyting1.html#K11" target="_self">diff</a> <span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="funcop_1.html#K6">.:</a> <span class="p1">(<span class="default"><a href="heyting1.html#K8" target="_self">pseudo_compl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><a href="heyting1.html#K11" target="_self">diff</a> <span class="p3">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p4">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:48"/><i><font color="Green">E45</font></i>: 
 <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="normform.html#T86">NORMFORM:86</a></i>;<br/>
<a NAME="E2:48"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="heyting1.html#K10" target="_self">SUB</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T1">ZFMISC_1:1</a></i>;<br/>
<a NAME="E3:48"/><i><font color="Green">E47</font></i>: 
 <a href="heyting1.html#K3" target="_self">@</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="heyting1.html#E1:48"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">H<sub>1</sub></font>(<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#T27">LATTICE2:27</a>, <a class="ref" href="lattice2.html#T29">LATTICE2:29</a></i>;<br/>
<i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><a href="heyting1.html#K11" target="_self">diff</a> <span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="subset_1.html#K6">\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="heyting1.html#E1:48"><i><font color="Green">E45</font></i></a>, <a class="ref" href="heyting1.html#D11" target="_self">Def11</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="normform.html#T86">NORMFORM:86</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K8">Normal_forms_on</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="heyting1.html#T22" target="_self">Th22</a></i>;<br/>
<i><font color="Green">E50</font></i>:  <a href="heyting1.html#K3" target="_self">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K8" target="_self">pseudo_compl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="heyting1.html#K8" target="_self">pseudo_compl</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="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><a href="heyting1.html#K6" target="_self">-</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <span class="p3">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p4">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="heyting1.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="normform.html#K9">mi</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="heyting1.html#E3:48"><i><font color="Green">E47</font></i></a>, <a class="ref" href="heyting1.html#T18" target="_self">Th18</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="normform.html#T66">NORMFORM:66</a></i>
;<br/>
<i><font color="Green">E51</font></i>:  <a href="heyting1.html#K3" target="_self">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K7">[:]</a> <span class="p3">(<span class="default"><a href="heyting1.html#K11" target="_self">diff</a> <span class="p4">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p5">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p4">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K7">[:]</a> <span class="p2">(<span class="default"><a href="heyting1.html#K11" target="_self">diff</a> <span class="p3">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p4">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="heyting1.html#E5:48"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funcop_1.html#T60">FUNCOP_1:60</a></i>
<br/>.= 
 <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <span class="p3">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p4">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="heyting1.html#K7" target="_self">=&gt;&gt;</a> <span class="p2">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <span class="p3">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p4">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="heyting1.html#D9" target="_self">Def9</a></i>
<br/>.= 
 <a href="normform.html#K9">mi</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="heyting1.html#E3:48"><i><font color="Green">E47</font></i></a>, <a class="ref" href="heyting1.html#T19" target="_self">Th19</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="normform.html#T66">NORMFORM:66</a></i>
;<br/>
<b>thus </b> <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="filter_0.html#T38">FILTER_0:38</a></i>
<br/>.= 
 <a href="lattice2.html#K2">FinJoin</a> <span class="p1">(<span class="default"><a href="heyting1.html#K10" target="_self">SUB</a> <span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="funcop_1.html#K6">.:</a> <span class="p2">(<span class="default"><a href="heyting1.html#K8" target="_self">pseudo_compl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K7">[:]</a> <span class="p3">(<span class="default"><a href="heyting1.html#K11" target="_self">diff</a> <span class="p4">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p5">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p4">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="heyting1.html#T33" target="_self">Th33</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="heyting1.html#K10" target="_self">SUB</a> <span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="funcop_1.html#K6">.:</a> <span class="p2">(<span class="default"><a href="heyting1.html#K8" target="_self">pseudo_compl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K7">[:]</a> <span class="p3">(<span class="default"><a href="heyting1.html#K11" target="_self">diff</a> <span class="p4">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p5">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p4">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattice2.html#D3">LATTICE2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="funcop_1.html#K6">.:</a> <span class="p2">(<span class="default"><a href="heyting1.html#K8" target="_self">pseudo_compl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K7">[:]</a> <span class="p3">(<span class="default"><a href="heyting1.html#K11" target="_self">diff</a> <span class="p4">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p5">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p4">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p2">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="heyting1.html#E2:48"><i><font color="Green">E46</font></i></a>, <a class="txt" href="heyting1.html#E4:48"><i><font color="Green">E48</font></i></a>, <a class="ref" href="setwiseo.html#T26">SETWISEO:26</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<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="heyting1.html#K8" target="_self">pseudo_compl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="heyting1.html#K9" target="_self">StrongImpl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funcop_1.html#K7">[:]</a> <span class="p3">(<span class="default"><a href="heyting1.html#K11" target="_self">diff</a> <span class="p4">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p5">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p4">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <span class="p3">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T48">FUNCOP_1:48</a></i>
<br/>.= 
 <a href="normform.html#K9">mi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="normform.html#K10">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="heyting1.html#E7:48"><i><font color="Green">E50</font></i></a>, <a class="txt" href="heyting1.html#E8:48"><i><font color="Green">E51</font></i></a>, <a class="txt" href="heyting1.html#E8"><i><font color="Green">Lemma5</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="heyting1.html#T7" target="_self">Th7</a></i>
;<br/>


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